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*Table of contents : Fundamental Group of the Affine Line in Positive Characteristic......Page 9Impact of geometry of the boundary on the positive solutions of a semilinear Neumann problem with Critical nonlinearity......Page 40Sur la cohomologie de certains espaces de modules de fibrés vectoriels......Page 52Some quantum analogues of solvable Lie groups......Page 57Compact complex manifolds whose tangent bundles satisfy numerical effectivity properties......Page 85Algebraic Representations of Reductive Groups over Local Fields......Page 109Poncelet Polygons and the Painlevé Equations......Page 185Scalar conservation laws with boundary condition......Page 226Bases for Quantum Demazure modules-I......Page 238An Appendix to Bases for Quantum Demazure modules-I......Page 265Moduli Spaces of Abelian Surfaces with Isogeny......Page 270Instantons and Parabolic Sheaves......Page 292Numerically Effective line bundles which are not ample......Page 319Moduli of logarithmic connections......Page 323The Borel-Weil theorem and the Feynman path integral......Page 325Geometric Super-rigidity......Page 352*

Geometry and Analysis

TATA INSTITUTE OF FUNDAMENTAL RESEARCH STUDIES IN MATHEMATICS General Editor : S. RAMANAN 1. M. HervE: SEVERAL COMPLEX VARIABLES 2. M. F. Atiyah and others : DIFFERENTIAL ANALYSIS 3. B. Malgrange : IDELAS OF DIFFERENTIABLE FUNCTIONS 4. S. S. Abhyankar and others : ALGEBRAIC GEOMETRY 5. D. Mumford: ABELIAN VARIETIES 6. L.Schwartz: RANDON MESAURES ON ARBITRARY TOPLOGICAL SPACES AND CYLINDIRICAL MEASURES 7. W. L.Bail Jr. and others: DISCRETE SUBGROUPS OF LIE GROUPS AND APPLICATIONS TO MODULI 8. C.P. RAMANUJAM -A TRIBUTE 9. C.L. Siegel: ADVANCED ANALYTIC NUMBER THEORY 10. S. Gelbert and others: AUTOMORPHIC FORMS, REPRESENTATION THEORY AND ARITHMETIC 11. M. F. Atiyah and others: VECTOR BUNDLES ON ALGEBRAIC VARIETIES 12. R.Askey and others: NUMBER THEORY AND RELATED TOPICS 13. S.S. Abhyankar and others: GEOMETRY AND ANALYSIS

GEOMETRY AND ANALYSIS

Papers presented at the Bombay Colloquium 1992, by

ABHYANKAR ADIMURTHI BEAUVILLE BIRKENHAKE DE CONCINI DEMAILLY HABOUSH HITCHIN JOSEPH KAC LAKSHMIBAI LANGE MARUYAMA MEHTA NITSURE OKAMOTO PROCESI SIU SUBRAMANIAN

Published for the TATA INSTITUTE OF FUNDAMENTAL RESEARCH, BOMBAY

OXFORD UNIVERSITY PRESS 1995

Oxford University Press, Walton Street, Oxford OX2 6DP OXFORD NEW YORK ATHENS AUCKLAND BANGKOK BOMBAY CALCUTTA CAPE TOWN DAR ES ASLAAM DELHI FLORENCE HONG KONG ISTANBUL KARACHI KUALA LUMPUR MADRAS MADRID MELBOURNE MEXICO CITY NAIROBI PARIS SINGAPORE TAIPEI TOKYO TORONTO and associates in BERLIN IBADAN Oxford is a trademark of Oxford University Press

© Tata Institute of Fundamental Research, 1995 ISBN 0 19 563740 2

Printed at Konam Printers, Bombay ´ Published by Neil OBrien, Oxford University Press, Oxford House, Apollo Bunder, Bombay 400 001.

INTERNATIONAL COLLOQUIUM ON GEOMETRY AND ANALYSIS BOMBAY, 6-14 JANUARY 1992 REPORT AN INTERNATIONAL COLLOQUIUM on ‘Geometry and Analysis’ was held at the Tata Institute of Fundamental Research, Bombay from January 6 to January 14, 1992. Professors M. S. Narasimhan and C.S. Seshadri turned sixty at about this time. In view of the crucial role both of them played in the evolutions of the School of Mathematics as a centre of excellence, it was considered appropriate to devote the Colloquium to recent developments in areas of geometry and analysis close to their research interests, and to felicitate them on this occasion. The range of topics dealt with included vector bundles, moduli theory, complex geometry, algebraic and quantum groups, and differential equations. The Colloquium was co-sponsored by the International Mathematical Union and the Tata Institute of Fundamental Research, and was financially supported by them and the Sir Dorabji Tata Trust. The Organizing Committee for the Colloquium consisted of Professors Kashmibai, David Mumford, Gopal Prasad, R. Parthasarathy (Chairman), M.S. Raghunathan, T. R. Ramadas, S. Ramanan and R. R. Simha. The International Mathematical Union was represented by Professor Mumford. The following mathematicians gave one-hour addresses at the Colloquium: S.S. Abhyankar, A. Adimurthy, A. Beauville, F.A. Bogomolov, C. de Concini, J.P. Demailly, W. J. Haboush, G. Harder, A. Hirschowitz, N. J. Hitchin, S. P. Inamber, K. T. Joseph, G. R. Kempf, V. Lakshmibai, H. Lange, M. Maruyama, D. Mumford, M. P. Murthy, N. Nitsure, M. V. Nori, K. Okamoto, C. Procesi, N. Raghavendra, S. Ramanan, Y. T. Siu, V. Srinivas, S. Subramanian, G. Trautmann. Besides the members of the School of Mathematics of the Tata Institute, mathematicians from universities and educational institutions in India and abroad were also invited to attend the Colloquium.

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The social programme for the Colloquium included a Tea Party on January 9, a documentary film on 6 January, a Carnatic Flute Recital on January 8, a Hindustani Vocal Recital on January 11, a Kathakali Dance on January 13, a Dinner at Gallops Restaurant, Mahalaxmi on January 12, an Excursion to Elephanta Caves on January 14, and a Farewell Dinner Party on January 14, 1992.

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Contents 1

Fundamental Group of the Affine Line in Positive Characteristic

1

2

Impact of geometry of the boundary on the positive solutions of a semilinear Neumann problem with Critical nonlinearity 32

3

Sur la cohomologie de certains espaces de modules de fibr´es vectoriels 44

4

Some quantum analogues of solvable Lie groups

5

Compact complex manifolds whose tangent bundles satisfy numerical effectivity properties 77

6

Algebraic Representations of Reductive Groups over Local Fields 101

7

Poncelet Polygons and the Painlev´e Equations

177

8

Scalar conservation laws with boundary condition

218

9

Bases for Quantum Demazure modules-I

230

49

10 An Appendix to Bases for Quantum Demazure modules-I

257

11 Moduli Spaces of Abelian Surfaces with Isogeny

262

vii

12 Instantons and Parabolic Sheaves

284

13 Numerically Effective line bundles which are not ample

311

14 Moduli of logarithmic connections

315

15 The Borel-Weil theorem and the Feynman path integral

317

16 Geometric Super-rigidity

344

viii

Fundamental Group of the Affine Line in Positive Characteristic Shreeram S. Abhyankar*

1 Introduction I have known both Narasimhana and Seshadri since 1958 when I had 1 a nice meal with them at the Student Cafeteria in Cit´e Universitaire in Paris. So I am very pleased to be here to wish them a Happy Sixtieth Birthday. My association with the Tata Institute gore back even further to 1949-1951 when, as a college student, I used to attend the lectures of M. H. Stone and K. Chandraseksharan, first in Pedder Road and then at the Yacht Club. Then in the last many years I have visited the Tata Institute numerous times. So this Conference is a nostalgic homecoming to me. To enter into the subject of Fundamental Groups, let me, as usual, make a.

2 High-School Beginning So consider a polynomial f “ f pYq “ Y n ` a1 Y n´1 ` ¨ ¨ ¨ ` an with coefficients a1 , . . . , an in some field K; for example, K could be the field of rational numbers. We want to solve the equation f “ 0, i.e., we want to find the roots of f . Assume that f is irreducible and has no multiple roots. Suppose somehow we found a root y1 of f . Then to make * Invited Lecture delivered on 8 January 1992 at the International Colloquium on Geometry and Analysis in TIFR in Bombay. This work was partly supported by NSF Grant DMS-91-01424

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the problem of finding the other roots easier, we achieve a decrease in degree by “throwing away” the root y1 to get f1 “ f1 pYq “ 2

f pYq “ Y n´1 ` b1 Y n´2 ` ¨ ¨ ¨ ` bn´1 . Y ´ y1

If f1 pYq is also irreducible and if somehow we found a root y2 of f1 , then “throwing away” y2 we get f2 “ f2 Y “

f1 pYq “ Y n´2 ` c1 Y n´3 ` ¨ ¨ ¨ ` cn´2 . Y ´ y2

Note that the coefficients b1 . . . , bn´1 of f1 do involve y1 and hence they are note in K, but they are in Kpy1 q. So, although we assumed f to be irreducible in KrYs, when we said “if f1 is irreducible,” we clearly meant “if f1 is irreducible in Kpy1 qrYs”. Likewise, irreducibility of f2 refers to its irreducibility in Kpy1 , y2 qrYs. And so on. In this way we get a sequence of polynomials f1 , f2 , . . . , fm of degrees n´1, n´2, . . . , n´m in Y with coefficients in Kpy1 q, Kpy1 , y2 q, . . . , Kpy1 , . . . , ym q, where fi is irreducible in Kpy1 , . . . , yi qrYs for i “ 1, 2, . . . , m ´ 1. If fm is reducible in Kpyp 1q, . . . , ym qrYs then we stop, otherwise we proceed to get fm`1 , and so on. Now we may ask the following. Question: Given any positive integers m ă n, doers there exist an irreducible polynomial f of degree n in Y with coeffcients in some field K such that the above sequence terminates exactly after m steps, i.e., such that f1 , f2 , . . . , fm´1 are irreducible but fm is reducible? I presume that most of us, when asked to respond quickly, might say: “Yes, but foe large m and n it would be time consuming to write down concrete examples”. However, the SURPRISE OF THE CENTURY is that the ANSWER is NO. More precisely, it turns out that f1 , f2 , f3 , f4 , f5 irreducible ñ f6 , f7 , . . . , fn´3 irreducible. In other words, if f1 , . . . , f5 are irreducible then f1 , . . . , fn´1 are all irreducible except the fn´2 , which is a quadratic, may or may not be irreducible. This answers the case of m ě 6. Going down the line to m ď 5 and assuming m ă n ´ 2, for the case of m “ 5 we have that 2.1

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Fundamental Group of the Affine Line in Positive Characteristic

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2.2 f1 , f2 , f3 , f4 irreducible but f5 reducible ñ n “ 24 or 12 and for the case of m “ 4 we have that 2.3

f1 , f2 , f3 irreducible but f4 reducible ñ n “ 23 or 11. Going further down the line, for the case of m “ 3 we have that

f1 , f2 irreducible but f3 reducible ñ Refined FT of Proj Geom 3 i.e., if f1 , f2 are irreducible but f3 is reducible,then there are only a few possibilities and they are suggested by the Fundamental Theorem of Projective Geometry, which briefly says that “the underlying division ring of a synthetically defined desarguestion projective plane is a field in and only if any three point of a projective line can be mapped to any other three points of that projective line by a unique projectivity.” Going still further down the line for the case of m “ 2 we have that

2.4

f1 irreducible but f2 reducible ñ known but too long i.e., if f1 is irreducible but f2 is reducible, the answer is known but the list of possibilities is too long to write down here. Finally, for the case of m “ 1 we have that 2.5

f1 has exactly two irreducible factors ñ Pathol proj Geom ` Stat i.e., if f1 has exactly two irreducible factors, then again a complete answer is known, which depends on Pathological Projective Geometry and Block Designs from Statistics! Here I am reminded of the beautiful course on Projective Geometry which I took from Zariski (in 1951 at Harvard), and in which I learnt the Fundamental Theorem mentioned in (2.4). At the end of that course, Zariski said to me that “Projective geometry is a beautiful dead subject, so don’t try to do research in it” by which he implied that the ongoing research in tha subject at that time was rather pathological and dealt with non- desarguesian planes and such. But in the intervening thirty or forty years, this “pathological” has made great strides in the hands of pioneers from R. C. Bose [21] and S. S. Shrikhande [58] to P. Dembowski [29] and D. G. Higman [34], and has led to a complete classification of Rank 3 groups, which 2.6

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from our view-point of the theory of equations is synonymous to case (2.6). So realizing how even a great man like Zariski could be wrong occasinally, I have learnt to drop one of my numerous prejudices, namely my prejudice against Statistics. Note that a permutation group is said to be transitive if any point (of the permuted set) can be sent to any other, via a permutation in the group. Likewise, a permutation group is m-fold transitive (briefly: m-fold transitive) if any m points can be sent to any other m points, via a permutation in that group. DoublyT ransitive “ 2 ´ transitive, T riplyT ransitive “ 3 ´ transitive, and so on. By the one point stabilizer of a transitive permutation group we mean the subgroup consisting of those permutations which keep a certain point fixed; the orbits of that subgroup are the minimal subsets of the permuted set which are mapped to themselves by every permutation in that subgroup; of the nontrivial orbits are called the sub degrees of the group, so that the numbers of sub degress is one less than than rank. Thus a Rank 3 group is transitive permutation group whose one point stabilizer has three orbits; the lengths of the two nontrivial orbits are the sub degrees. Needless to say that a Rank 2 group is nothing but a Doubly Transitive permutation group. At any rate, in case (2.6), the degrees of the two irreducible factors of f1 correspond to the sub degrees of the relevant Rank 3 group. Now CR3(= the Classification Theorem of Rank 3 groups) implies that very few pairs of integres can be the sub degrees of Rank 3 groups, very few nonisomorphic Rank groups can have the same sub degrees; see KantorLiebler [42] and Liebeck [44]. Hence (2.6) says that if f1 has exactly two irreducible factors then their degrees (and hence also n) can have only certain very selective values. Here, by the relevant group we mean the Galois group of f over K, which we donate by Galp f, Kq and which, following Galois, we define as the group of those permutations of the roots y1 . . . , yn which retain all the polynomial relations between them wiht coefficients in K. This definition makes sense without f being irreducible but still assuming f to have no multiple roots. Now our assumptio of f being irreducible is equivalent to assuming that Galp f, Kq is transitive. Likewise, f1 , . . . , fm´1 are irreducible iff Galp f, Kq in m-transitive. Moreover, as 4

Fundamental Group of the Affine Line in Positive Characteristic

5

already indicated, f1 has exactly two irreducible factors iff Galp f, Kq has rank 3. To match this definition of Galois with the modern definition, let L be the splitting field of f over K, i.e., L “ Kpy1 . . . , yn q. Then according to the modern definition, the Galois group of L over k, denoted by GalpL, Kq, is defined to be the group of all automorphisms of L which keep K point wise fixed. Considering Galp f, Kq as a permutations group of the subscripts 1, . . . , n of y1 , . . . , yn for every τ P GalpL, Kq we have a unique σ P Galp f, Kq such that τpyi q “ yσpiq for 1 ď i ď n. Mow we get an isomorphism of GalpL, Kq onto Galp f, Kq by sending each τ to the corresponding σ. Having sufficiently discussed case (2.6), let us note that (2.5) is equivalent to CDT (=Classification Theorem of Doubly Transitive permutation groups) fr which we manu refer to Cameron [22] and Kantor [41]. At any rate, CDT implies that if f1 is irreducible but f2 is reducible then we must have: either n “ q for some prime power q, or n “ pql ´ 1q{pq ´ 1q for some integer l ą 1 and some prime power q, or n “ 22l´1 ´ 2l´1 for some integer l ą 2, or n “ 15, or n “ 176, or n “ 276. Likewise (2.4) is equivalent to CTT(= Classification of Triply Transitive permutation groups) which is subsumed in CDT, and 5 as a consequence of it we can say that if f1 , f2 are irreducible but f3 is reducible then we must have: either n “ 2l for some positive integer l, or n ´ q ` 1 for some prime powder q, or n “ 22. Similarly, (2.3) is equivalent to CQT (= Classification of Quadruply Transitive permutation groups) which is subsumed in CTT, and as a consequence of it we can say that if f1 , f2 , f3 are irreducible but f4 is reducible then we must have: eithee n “ 23 and Galp f, Kq “ M23 or n “ 11 and Galp f, Kq “ M11 , where M stands for Mathieu. Likewise, (2.2) is equivalent to CFT(= Classification of Fivefold Transitive permutation groups) which is subsumed in CQT, and as a consequence of it we can say that if f1 , f2 , f3 , f4 are irreducible but f5 is reducible then we must have: either n “ 24 and Galp f, Kq “ M24 and Galp f, Kq “ M12 . Note that, M24 and M24 are the only 5-told but not 6-fold transitive permutation groups other than the symmetric group S 5 (i.e., the group of all permutations on 5 letters) and the alternating group A7 (i.e., the sub-group of s7 consisting of all even permutations). Moreover, M23 and 5

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Shreeram S. Abhyankar

M11 are the respective one point stabilizers of M24 and M12 and they are the only 4-fold but not 5-fold transitive permutation groups other than s4 and A6 . Here the subscript denotes the degree, i.e., the number of letters being permuted. The four groups M24 , M23 , M12 , M11 , were constructed by Mathieu [46] in 1861 as examples of highly transitive permutation groups. But the fact that they are the only 4-transitive permutation groups others than the symmetric groups and the alternating groups, was proved only in 1981 when CDT, and hence also CTT, CQT, CFT and CST, were deduced from CT(= Classification Theorem of finite simple groups); see Cameron [23] and Cameron-Cannon [24]. Recall that a group is simple if it has no nonidentity normal subgroup other than itself; it turns out that the five Mathieu groups M24 , M23 , M22 , M12 , M11 and M22 is the point stabilizer of M23 , are all simple. Now CST refers to the Classification Theorem of Sixfold Transitive permutation groups, according to which the symmetric groups and the alternating groups are the only 6-transitive permutation groups; note that S m is m-transitive but not pm ` 1q-transitive, whereas Am is pm ´ 2q-transitive but not pm ´ 1qtransitive for m ě 3. In (2.2) to (2.6) we had assumed M ă n ´ 2 to avoid including the symmetric and alternating groups; dropping this assumption, (2.1) is equivalent to CST with the clarification that, under the assumption of (2.1), the quadratic fn´2 is irreducible or reducible according as Galp f, Kq “ sn or An . We have already hinted that CR3 was also deduced as a consequence of CT; Liebeck [44]. The proof of CT itself was completed in 1980 (see Gorenstein [32]) with staggering statistics: 30 years; 100 authors; 500 papers; 15,000 pages! Add some more pages for CDT and CR3 and so on. All we have done above is to translate this group theory into te language of theory of equations where K is ANY field. So are still talking High-School? Not really, unless we admint CT into High-School!

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Fundamental Group of the Affine Line in Positive Characteristic

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3 Galois Summarizing, to compute the Galois group Galp f, Kq, say when K is the field kpXq of univariate rational functions over an algebraically closed ground field k, by throwing away roots and using some algebraic geometry we find some multi-transitivity and other properties of the Galois groups and fedd there into the group theory machine. Out comes a list of possible groups. Reverting to algebraic geometry, sometimes augmented by High-School manipulations, we successively eliminate various members from that list until, hopefully, one is-left. That then is the answer. I say hopefully because we would have a contradiction in which the ultimate reality (Brahman) is described by Neti Neti, not this, not that. If you practice pure Advaita, then nothing is left, which is too austere. So we fall back on the kinder Dvaita according to which the unique God remains.

4 Riemann and Dedekind In case K “ CpXq and ai “ ai pXq P CrXs for 1 ď i ď n, where C is the field of comples numbers, following Riemann [53] we can consider the monodromy group of f thus. Fix a nondiscriminant point µ, i.e., value mu P C of X for which the equation f “ 0 has n distinct roots. Then, say by the Implicit Function Theorem, we can solve the equation the equation f “ 0 near µ, getting n analytic solutions η1 pXq, . . . , ηn pXq near µ. To find out how there solutions are intertwined, mark a finite number of values α1 , . . . , αw of X which are different from µ but include all the discriminant points, and let Cw be the complex X-plane minus these w points. Now by making analytic continuations along any closed path Γ in Cw starting and ending at µ so that ηi continues into η j with Γ1 piq “ j for 1 ď i ď n. As Γ varies over all closed paths in Cw starting and ending at µ, the permutations Γ1 span a subgroup of S n called the monodromy group of f which we denote bye Mp f q.

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Shreeram S. Abhyankar

By identifying the analytic solutions η1 , . . . , ηn with the algebraic roots y1 , . . . , yn , the monodromy group Mp f q gets identified with the Galois group Galp f, CpXqq, and so these two groups are certainly isomorphic as permutation groups. To get generators for Mp f q, given any α P Cw , let Γα be the path in Cw consisting of a line segment from µ to a point very near α followed by a small circle around α and then back to µ along the said line segment. Let us write the corresponding permutation Γ1α as a product of disjoint cycles, and let e1 , . . . , eh be the lengths of these cycles. To get a tie-up between these Riemannian considerations and the thought of Dedekind [28], let v be the valuation of CpXq corresponding to α, i.e., vpgq is the order of zero at alpha for every g P CrXs. Then, as remarked in my 1957 paper [3], the cycle lengths e1 , . . . , en coincide with the ramification exponents of the various extensions of v to the root field CpXqpy1 q, and their LCM equals the ramification exponent of any extension of v to the splitting field CpXqpy1 , . . . , yn q. In particular, Γ1α is the identity permutation iff α is not a branch point, i.e., if and only if the ramification exponents of the various extensions of v to the root field CpXqpy1 q (or equivalently to the splitting field CpXqpy1 , . . . , yn q) are all 1. At any rate, a branch point is always a dicriminant point but not conversely. Indeed, the difference between the two is succinctly expressed by Dedekind’s Theorem according to which the ideal generated by the Y-derivative of f equals the products of the different and the conductor. In this connection you may refer to pages 423 and 438 of any my Monthly Article[5] which costitutes some of my Ramblings in the woods of algebraic geometry. You may also refer to pages 65 and 169 of my recent book [6] for Scientists and Engineers inti which these Ramblings have now been expanded. Having given a tie-up between the ideas of Riemanna and Dedekind (both of whom wre pupils of Gauss) concerning branch points, ramification exponents, and so on, it is time to say that these things actually go back to Newton [47]. For an excellent discussion of the seventeenth century work of Newton on this matter, see pages 373-397 of Part II of the 1886 Textbook of Algebra by Chrystal [27]. For years having recommended Chrystal as the best book to learn algebra from, from time to 8

Fundamental Group of the Affine Line in Positive Characteristic

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time I decide to take my own advice a wealth of information it contains! At any rate, a` m la Newton, we can use fractional power series in X to factor f into linear factors śh in Y, and then combine conjugacy classes to get a factorization f “ i“1 φi where φi is an irreducible ploynomail of degree ei in Y whose coefficients are power series in X ´α. If the field C were not algebraically closed then the degree of φi would be ei fi with fi being certain “residuce degrees” and we would get the famous formula σhi“1 ei fi “ n of Dedekind- Domain Theory. See Lectures 12 and 21 of Scientists [6]. Geometrically speaking, i.e., following the ideas of Max Noether [48], if we consider the curve f “ 0 in the discriminant points cor- 8 respond to vertical lines which meet the curve in less than n point, the branch points correspond to vertical tangents, and the “ conductor points” are the singularities. See figure 9 on page 429 of Ramblings [5]. Getting back to finding generators for Mp f q, with the refinement of discriminant points into branch points and conductor points in hand, it suffices to stipulate that α1 , . . . , αw inculdue all the branch points rathaer than all the discrimant points. Now bye choosing the base point µ suitably, we may asume that the line from mu to α1 , . . . , αw do not meet each other except at µ. Now it will turn out that the permutations Γ1α1 , . . . , Γ1αw generate Mp f q. This follows from the Monodromy Theorem together with the fact that the (topological) fundamental group π1 pCw q of Cw (also called the Poincar`e group of Cw ) is the free group Fw on w generators. Briefly speaking, the monodromy Theorem says that two paths, which can be continuously deformed into each other, give rise to the same analytic continuations. The fundamental group itself may heuristically be described as that incarnation of the monodromy group which works for all functions whose branch points are amongest α1 , . . . , αw . More precisely, π1 qpCw q consists of the equivalent means they can be continuously deformed inti each other. Now the (equivalence classes of the) paths Γα1 , . . . , Γαw are free generators of π1 pCw q, and we have an obvious epimorphism of π1 pCw q “ Fw onto Mp f q and hence the permutations Γα1 , . . . , Γαw generate Mp f q; for relevant picture etc., you may see pages 442-443 of Ramblings [5] or pages 171-172 of Scientists [6]. So the curve f “ 0, or equivalently the Galois extension 9

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L “ CpXqpy1 , . . . , yn q, is an ramified covering of Cw , and tha Galois Group GalpL, CpXqq “ Gap f, CpXqq is generated by w generators. Surprisingly, to this day there is no algebraic proof of this algebraic fact. The Riemann Existence Theorem says that conversely, every finite homomorphic image of π1 pCw q can be realized as Mp f q for some f . Thus be defining the algebraic fundamental growp πA pCw q as the set of all finite groups which are the Galois groups of finite unramified coverings of Cw , we can say that πA pCw q coincides with the set of all finite groups generated by w generators. Needless to say that, a fortiori, there is no algebraic proof of the converse part of this algebraic fact either. Now, in the complex pX, Yq-plane, f “ 0 is a curve Cg of some genus g, i.e., if from Cg we delete a finite number of points including all its singularities, then what we get is homeomorphic to a sphere with g handles minus a finite number of points. For any nonnegative integer w, let Cg,w be obtained by adding to Cg its points at infinity, then desingularizing it, and finally removing w ` 1 points from the desingularized verison. Then Cg,w is homomorphic to a sphere with g handles minus w ` 1 points, and hence it can be seen that π1 pCg,w q “ F2g`w ; for instance see the excellent topology book of Seifert and Threlfall [54]. The above monodromy and existence considerations generalize fromn the genus zero case to the case of general g, and we get the result that the algebraic fundamental group πA pCg,w q coincides with the set of all finite groups generated be 2g ` w generators, where πA pCg,w q is defined to be the set of all finite groups which are the Galois groups of finite unramified coverings of Cg,w .

5 Chrystal and Forsyth Just as Chrystal excels in explaining Newtonian (and Eulerian) ideas, Forsyth’s 1918 book on Function Theory [31] is highly recommended for getting a good insight into Riemannian ideas. Thus it was by absorvbing parts of Forsyth that, in my recent papers [8] and [10], I could algebracize some of the monodromy considerations to formulate certain “Cycle Lemmas” which say that under such and sucn conditions the Galois group contains permutations having such and such cycle structure. 10

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11

Now the Rirmann Existence Theorem was only surmised be Riemann [52] by appealing to the Principle of his teacher Dirichlet which, after Weierstrass Criticism was put on firmee ground by Hilbert in 1904 [35]. In the meantime another classical treatment of the Riemann Existence Theorem was carried out culminating in the Klein-Poincar´e-Koebe theory of automorphic functions, for which again Forsyth’s book is a good source. A modern treatment of the Riemann Existence Theorem using coherent analytic sheaves was finally given by Serre in his famous GAGA paper [55] of 1956.

6 Serre Given any algebraically closed ground field k of any nonzero characteristic p, in my 1957 paper [3], all this led me to define and algebraic fundamental group πA pCg,w q of Cg,w “ Cg minus w ` 1 points, where w is a nonnegative integer and Cg is a nonsigular projetive curve of genus g over k, to be the set of all finite groups which can be realized as Galois groups of finite unramified coverings of Cg,w . In tha paper, I went on to conjecture that πA pCg,w q coincides with the set of all finite groups G for which G{ppGq is generated by 2g`w generators, where ppGq is the subgroup of G generated by all its p-Sylow subgroups. The g “ w “ 0 case of this conjecture, which may be called the quasi p-group conjecuture, says that for the affine line Lk over k we have πA pLk q “ Qppq where Qppq denotes the set of all quasi p-groups, i.e., finite groups which are 10 generated by their p-Sylow subgroups. It may be noted every finite simple group whose order is divisible bey p is obviously a quasi p-group. Hence in particular the alternating group An is a quasi p-group whenever either n ě p ą 2 or n ´ 3 ě p “ 2. Likewise the symmetric group S n is a quasi p-group provided n ě p “ 2. In support of the quasi p-group conjecture,in the 1957 paper. I wrote down several equations giving unramified coveing of the affine line Lk and suggested that their Galois groups be computed. This included the equation F n,q,s,a “ 0 with F n,q,s,a “ Y n ´ aX s Y t ` 1 and 11

n“q`t

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where 0 ‰ a P k and q is a positive power of p and s and t are positive integers with t ı 0ppq, and we want to compute its Galois group Gn,q,s,q “ GalpF n,q,s,q , kpXqq. By using a tiny amount of the information contained in the above equation, I showed that πA pLk q contains many unasolvable groups, and indeed by taking homomorphic imagtes of subgroups of members of πA pLk q we get all finite groups; see Result 4 and Remark 6 on pages 841-842 of [3]. This was somewhat of a surpise because the comples affine line is simply connected, and although πA pLk q was known to contain p-cyclic groups (so called Artin-Schreier equations), it was felt that perhaps it does not contain much more. This feeling, which turned out to be wrong, might have been based on the facts that Lk is a “commutative group variety” and the fundamental group of a topological group is always abelian; see Proposition 7 on page 54 of Chevalley [26]. To algebracize the fact that the comples affine line is simply connected, be the genus formula we deduce that the affine line over an algebraically closed ground field of characteristic zero has no nontrivial unramified coverings. In our case of characteristic p, the same formula shows that every membed of πA pLk q is a squasi p-group; see Result 4 on page 841 of [3]. Originally I found the above equation F n,q,s,a “ 0 by taking a section of a surface which I had constructed in my 1955 Ph.D. Thesis [1] to show that jung’s classical method [40] of surface desigularization doed not work for nonzero charactheristic because the local fundamental group above a normal crossing of the branch locus need not be solvable, while in the comples case it is always abelian. This failure of Jung’s method led me ti devise more algoprithmic techniques fr desingularizing surfaces in nonzero characteristic, and this formed the positive part of my Ph.D. Thesis [2]. Soon after the 1957 paper, I wrote a series of articles [4] on “tame coverings” of higher dimensional algebraic varieties, and took note of Grothendieck [33] proving the “tame part” of the above conjecture which 11 says that the members of πA pCg,w q whose order is prime to p are exavtly all the finite gropus of order prime to p generated by 2g ` w generators. 12

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13

But after these two things,for a long time I forgot all about covering and fundamental groups. Then suddenly, after a lapse of nearly thirty years, Serre pulled me back into the game in October 1988 by writing to me a series of letters in which be briefly said: “I can now show that if t “ 1 then Gn,q,s,a “ PS Lp2, qq. Can you compute Gn,q,s,a for f ě 2? Also, can you find unramified An coverings of Lk ?” Strangely, the answers to both these questions turned out to be almost the same. Namely, with much prodding and prompting by Serre (hundred e-mails and a dozen s-mails=snail-mails) augmented by groups theory lessons first from Kantor and Feit and then Cameron and O’Nan, and by using the method of throwing away roots, CT in the guise of CDT, the Cycle Lemmas, the Jordan-Marggraff Theorems on limits of transitivity (see Jordan [39] and Marggraff [45] or Wielandt [60]), and finally some High-School type factorizations, in the papers [8] to [10] I proved that: t “ 1 ñ Gn,q,s,a “ PS Lp2, qq.

(6.1)

q “ p ą 2 ď t and pp, tq ‰ p7, 2q ñ Gn,q,s,a “ An .

q “ p ą 2 ď t and pp, tq “ p7, 2q ñ Gn,q,s,a “ PS Lp2, 8q.

q “ p “ 2 ñ Gn,q,s,a “ S n .

(6.2) (6.3) (6.4)

q “ p ą 2 ą and pp, tq ‰ p7, 2q ñ Gn,q,s,a “ An . p “ 2 ă q ă t ñ Gn,q,s,a “ An .

p “ 2 ă q “ 4and t “ 3pandn “ 7q ñ Gn,q,s,a “ PS Lp3, 2q. p “ 3 ă q “ 9and t “ 2pandn “ 11q ñ Gn,q,s,a “ M11 .

(6.5) (6.6) (6.7) (6.8)

Note that PS Lpm.qq “ S Lpm, qq{pscalarmatricesq where S Lpm, qq “ The group of all m by m matrices whose determinant is 1 and whose entries are in the field GFpqq of q elements. Now my proof of (6.1) uses the Zassenhaus-Feit-Suzuki Theorem which characterizes doubly transitive permutation groups for which no 3 points are fixed by a nonidentity permutation; see Zassenhaus [62], Feit [30] and Suzuki [59]. As Serre has remarked, his proof of (6.1) may be called a “descending” 13

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proof as opposed to my “ascending” proof. Serre’s proof may be found in his November 1990 letter to me which appears as an Appendix to my paper [8]. Actually, when [8] was already in press, Serre found that a proof somewhat similar to his was already given by Carlitz [25] in 1956. Throwing away one root of F n,q,s,a and then applying Abhyankar’s Lemma (see pages 181-186) of Part III of [4]) and deforming things conveniently, we get the monic polynomial F n,q,s,a,b,u of degree n ´ 1 in Y with coefficient in kpxq given by “ ‰ 1 F n,q,s,a,b,u “ t´2 pY ` tqt ´ Y t pY ` bqq ´ aX ´s Y u

with 0 ‰ b P k and positive integer u, n ´ 1. Now upon letting ˆ ˙ q´1 r “ pq ` tqLCM t, GCDpq ´ 1, q ` tq 1

1

and Gn,q,s,a,b,u “ GalpF n,q,s,a,b,u , kpXqq, in the papers [8] and [10] I also 1

proved that, in the following cases, F n,q,s,a,b,u “ 0 gives an unramified covering of Lk with the indicated Galois group: (6.11 ) b “ u “ t ą 2 ‰ q “ p and s ” 0pp ´ 1q and s ” 0ptq ñ Gn,q,s,a,b,u “ An´1 . 1

(6.21 ) b “ u “ t “ 2 and q “ p ‰ 7 and s ” 0pp ´ 1q ñ Gn,q,s,a,b,u “ An´1 . (6.31 ) If t “ 2 and q “ p ą 5 then u can be chosen so that 1 ă u ă pp ` 1q{2 and GCDpp ` 1, uq “ 1, and for any such u upon assuming b “ u{pu ´ 1q and s ” 0pupp ` 1 ´ uqq, we have 1 Gn,q,s,a,b,u “ An´1 . 1

(6.41 ) b “ u “ t and q “ p “ 2 and s ” 0ptq ñ Gn,q,s,a,b,u “ S n´1 . 1

(6.51 ) b “ u “ t ą q and p ą 2 and s ” 0prq ñ Gn,q,s,a,b,u “ An´1 . 1

(6.61 ) b “ u “ t ą q “ p “ 2 and s ” 0prq ñ Gn,q,s,a,b,u “ S n´1 . 14

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1

(6.71 ) b “ u “ t ą q ą p “ 2 and s ” 0prq ñ Gn,q,s,a,b,u “ An´1 . Another equation written down in the 1957 paper giving an unramified covering of Lk is Frn,t,s,a “ 0 where n, t, s are positive integers with t ă n ” 0ppq and GCDpn, tq “ 1 and s ” 0ptq

and Frn,t,s,a in the polynomial given by

Frn,t,s,a “ Y n ´ aY t ` X s with 0 ‰ a P k.

Again upon letting Frn,t,s,a “ GalpFrn,t,s,a , kpXqq, in the papers [8] and [10] I proved that: rn,t,s,a “ An . (6.1*) 1 ă t ă 4 and p ‰ 2 ñ G

rn,t,s,a “ An . (6.2*) 1 ă t ă n ´ 3 and p ‰ 2 ñ G

rn,t,s,a “ An . (6.3*) 1 ă t “ n ´ 3 and p ‰ 2 and 11 ‰ p ‰ 23 ñ G

rn,t,s,a “ An . (6.4*) 1 ă t ă 4 ă n and p “ 2 ñ G

rn,t,s,a “ An . (6.5*) 1 ă t ă n ´ 3 and p “ 2 ñ G

In Proposition 1 of the 1957 paper I discussed the polynomial h´1 ř Y hp`t ` aXY hp ` 1 ` ai Y ph´iqp with t ” 0ppq and 0 ‰ a P k and i“1

ai P k giving an unramified covering of Lk . The polynomial F studied 13 in (6.1) to (6.6) is the hp “ q and a1 “ . . . “ ah´1 “ 0 ace of this after “reciprocating” the roots and changing X to X s . Considering the p “ 2 “ h “ t ´ 1 and a “ a1 “ 1 case this we get the polynomial. F ˝ “ Y 7 ` xY 4 ` Y 2 ` 1 and it can be shown that: p6.1˝ q For p “ 2 the equation F ˝ “ 0 gives an unramified coveing of Lk with GalpF ˝ , kpXqq “ A7 . 15

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By throwing away a root of f ˝ and then invoking Abhyankar’s Lemma we obtain the polynomial F 1˝ “ Y 6 `X 27 Y 5 `x54 Y 4 `pX 18 `X 36 qY 3 `X 108 Y 2 `pX 90 `X 135 qY`X 162 and therefore by p6.1˝ q we see that: p6.2˝ q for p “ 2 the equation F 1˝ “ 0 gives an unramified covering of Lk with GalpF 1˝ , kpXqq “ A6 . Now it was Serre who first propted me to use CT in calculating the various Galois groups discsussed above. But after I had done this, agin it was Serre who groups discussed above. Bit after I had done this, agin it was Serre who prodded me to try to get around CT. So, as described in the papers [8] and [10], by traversing as suitable path in items (6.1) to (6.2˝ ) we get a complete equational proof of the following Facts without CT: Facts. (6.i) For all n ě p ą 2 we have An P πA pLk q . (6.ii) For all n ě p “ 2 we have S n P πA pLk q. (6.iii) For n ě p “ 2 a with 3 ‰ n ‰ 4 we have An P πA pLk q: (note that A3 and A4 are not quasi 2-groups). While attempting to circumvent CT, once I got a very amusing email from Serre saying ‘About the essential removal of CT from your An -determinations: what does essential mean? (Old story: a noble man had a statue of himself made be a well-known sculptor. The sculptor asked: do you want an equestrian statue or not? The noble man did not understand the word. He said: oh, yes, equestrian if you want, but not too much. . .). This is what I feel about non essential use of CT.” In any case, learnings and adopting (or adapting) all this group theory has certainly been very rejuvenating to me. To state CT very briefly: Z p (= the cyclic group of prime order p), An (excluding n ď 4), PSLpn` 1, qq (excluding n “ 1 and q ď 3) together with 15 other related and reincarnated infinite families, and the 26 sporadics including the 5 Mathieus is a complete list of finite simple groups; for details see Abhyankar [8] and Gorenstein [32]. 16

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7 Jacobson and Berlekamp Concerning items (6.6), (6.71 ), (6.4*) and (6.5*), when I said that I proved them in [8] and [10], what I actually meant was that I proved their weaker version asserting that the Galois group is the alternating group of the symmetric group, and then thanks to Jacobson’s Criterion,the symmetric group possibility was eliminated in my joint paper Ou and Sathaye [14]. What I am saying is that the classical criterion, according to which the Galois group of an equation is contained in the alternating two. A version of such a criterion which is valid for all characteristics including two was given by Jacobason in this Algebra books published in 1964 [37] and 1974 [38]; an essentially equivalent version may also be found in the 1976 paper [20] if Berlekamp with some preliminary work in his 1968 book [19]; both these criteria have a bearing on the Arf invariant of a quadratic form [18] which itself was inspired by some work of Witt [61]. This takes care of An coverings for characteristic two provided 6 ‰ n ‰ 7. This leaves us with the A6 and A7 coverings for characteristic two described in items (6.2˝ ) and (6.1˝ ). Again using the Jacobson’s Criterion, these are dealt with in my joint paper with Yie [17]. Thus, although Facts (6.i) and (6.ii) are indeed completely proved in my papers [8] and [10], but for Fact (6.iii) I was lucky to enjoy the active collaboration of my former (Sathaye) and present (Ou and Yie) students. Likewise, item (6.7) is not proved in my papers [8] to [10], but was communicated to me by Serre (e-mail of October 1991) and is included in my joint paper [17] with Yie. Similarly, item (6.8) is not in my papers [8] to [10], but is proved in my joint paper [15] with Popp and Seiler. In [8] I used polynomial Fr into the polynomial F and thereby get a proof of a stronger version of (6.1*) to (6.5*) without CT. Let us start by modifying the Third Irreducibility Lemma i Section 19 of [8] thus [in the proof of that lemma,once ξλ p1, Zq has been mistakenly printed as ξλp1,Zq ]:

17

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7.1 Let k be any field which need not be algebraically closed and whose characteristic chc k need not be positive. Let n ą t ą 1 be integers such that GCDpn, tq “ 1 and t ” 0pchc kq, and let ΩpZq be the monic polynomial of degree n´1 in Z with coefficients in kpYq obtained by putting rz ` Yqn ´ Y n s ´ rY n´t ` Y ´t s rpZ ` Yqt ´ Y t s . Z Then ΩpZq is irreducible in kpYqrZs. ΩpZq “

15

Proof. Since ı 0pchc, kq, upon letting

rpZ ` Yqn ´ Y n s ´ rpZ ` Yqt ´ Y t s and ηµ pY, Zq “ Z Z 1 by the proof of the above cited lemma we see that: ξλ1 pY, Zq and ηµ pY, Zq are homogeneous polynomials of degree λ1 “ n ´ 1 and µ “ t ´ 1 respectively, the polynomials ξλ1 1 p1, Zq and ηµ pY, Zq have no nonconstant common factor in krZs, and the polynomial ηµ pY, Zq has a has a nonconstant irreducible factor in krZs which does not divide ξλ1 1 p1, Zq and whose square does not divide ηµ pY, Zq. Upon letting ξλ1 1 pY, Zq “

Y t rpZ ` Yqn ´ Y n s ´ Y n rpZ ` Yqt ´ Y t s Z we see that ξλ pY, Zq is a homogeneous polynomail of degree λ “ n ` t ´ 1 and λp1, Zq “ ξλ1 1 pY, Zq ` ηµ p1, Zq and therefore: the polynomials ξλ pY, Zq and ηµ p1, Zq have no nonconstant commot fator in KrZs, and the polynomial ηµ p1, Zq has na nonconstant irreducible factor in krZs which does not divide ξλ p1, Zq and whose square does not divide ηµ p1, Zq. The proof of the Second Irreducibilitiy Lemma, of Section 19 of [8] clearly remains valid if only one of the polynomials ξλ pY, Zq and ηµ pY, Zq is assumed to be regular in Z, and in the present situation ηµ pY, Zq is obviously regular in A. Therefore by the said lemma, the polynomial “ξλ pY, Zq ` ηµ pY, Zq ‰is irreducible kpYqrZs. Obviously ΩpZq “ Y ´t ξλ pY, Zq ` ηµ pY, Zq and hence ΩpZq is irreducible in kpYqrZs. ξλ pY, Zq “

By using (7.1) we shall now prove: 18

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7.2 Let k be any field which need not be algebraically closed and whose characteristifc chc k need not be positive. Let 0 ‰ a P k and let n, t, s be positive integers such that 1 ă t ı 0pchc kq and 1 ă n ´ t ı 0pchc kq and GCDpn, tq “ 1. Then the polynomial ΦpYq “ Y n ´aX s Y t ` 1 is irreducible in kpXqrYs, its y-discriminant is nonzero, and for its Galois group we have: GalpΦpYq, kpXqq “ An or S n . Similarly, the polynomial ΨpYq “ Y n ´ aY t ` X x is irreducible in kpXqrYs, its Y-discriminant is nonzero, and for its Galois group we have: GalpΨpYq, kpXqq “ An or S n.

Proof. In view of the Basic Extension Principle and Corollaries (3.2) and (3.5) of the Substitutional Principle of Sections 19 of [8], without loss of generality we may assume that k is algebraically closed and a “ 1 “ s. Since Φ and Ψ are linear in X, they are irreducible. By the discriminant calculation in Section 20 of [8] we see that their Ydiscriminants are nonzero. As in the beginning of section 21 of [8] we 16 see that the valuation X “ 8 of kpXq{k splits into two valuations in the rood field of ΨpYq and their reduced ramification exponents are t and n´t. Now t and n´t are both nondivisible by chc k and GCDpt, n´tq “ 1, and hence by the Cycle Lemma of Section 19 of [8] we conclude that GalpΦpYq, kpXqq contains a t-cycle and an pn ´ tq-cycle. By throwing away a root of ΦpYq we get rΦpZ ` Yq ´ ΦpYqs {Z which equals ΩpZq because by solving ΦpYq “ 0 we get X “ Y n´t ` Y ´t . Consequently by (7.1) we Conclude that GalpΦpYqmnkpXqq is double transitive. Clearly either 1 ă t ă pn{2q or 1 ă n ´ t ă pn{2q, and hence by Marggraff’s Second Theorem as stated in Section 20 of [8] we get GalpΦpYq, kpXqq “ An or S n . By Corollaries (3.2) and (3.5) of the Substitutional Principle of section 19 of [8] it now follows that GalpY n ´ X t´n Y t ` 1, kpXqq “ An or S n . By multiplying throughout by xn , we obtain the polynomial Y n ´ Y t ` X n whose Galois group must be the same as the Galois group of Y n ´ X t´n Y t ` 1. Therefore GalpY n ´ Y t ` X n , kpXqq “ An or S n , and hence again by Corollaries (3.2) and (3.5) of the Substitutional Principle of Section 19 of [8] we conclude that GalpΨpYq, kpXqq “ An or S n . 19

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To get back to the polynomial Frn,t,s,a , let us return to the assumption of k being an algebraically closed field of nonzero characteristic p. Let 0 ‰ a P k and let n, t, s be positive integers with t ă n ” 0ppq and GCDpn, tq “ 1 and s ” 0ptq and recall that Frn,t,s,a “ 0 gives an unramified covering of Lk where Frn,t,s,a “ Y n ´ aY t ` X s

rn,t,s,a “ GalpwidetildeFn,t,s,a , and we want to consider the Galois group G kpXqq. Since every member of π4 pLk q is a quasi p-group and since S n in not a quasi p-group for pgeq3, in view of (2.28) of [14], by the Ψ case of (7.2) we get the following sharper version of (6.1*) to (6.5*): rn,t,s,a “ An . (7.1*) l ă t ă n ´ 1 ñ G

Just as the samall border values of t play a special role for the bar polynomial in (6.1) to (6.8), likewise the condition 1 ‰ t ‰ n ´ 1 in (7.1*) in not accidental as shown by the following four assertions: rn,t,s,a “ PS Lp2, 5q « A5 . (7.2*) 1 “ t “ n ´ 5 and p “ 2 ñ G

rn,t,s,a “ PS Lp2, 5q « A5 . (7.3*) 5 “ t “ n ´ 1 and p “ 2 ñ G

rn,t,s,a “ M p 11 « M11 . (7.4*) 1 “ t “ n ´ 11 and p “ 3 ñ G

17

rn,t,s,a “ pZ p qm . (7.5*) 1 “ t and n “ pm ñ G

Out of these four assertions, (7.2*) and (7.3*) may be found in my joint paper [17] with Yie, and (7.4*) may be found in my joint paper [15] with Popp and Seiler. If may be noted that PS Lp2, 5q and A5 are isomorphic as abstract groups but not as permutation groups. Likewise p 11 found by taking the image of the M11 found by taking the images M of the M11 under a noninner automorphism of M11 found by taking the image of the M11 found by taking the image of the M11 under a noninner automorphism of M12 ; see my paper [8] or volume III of the encyclopedic groups theory book [36] of Huppert and Blackburn. 20

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Concering assertion (7.5*), multiplying the roots by a suitable nonzero element of k we can reduce to the case of t “ 1 “ a and n “ pm and then, remembering that pZ p qm “ the m-fold direct product of the cyclic group Z p of order p “ the underlying additive group of GFppm q, our claim follows from the following remark: m (7.1**) If Y p ´ Y ` x is irreducible over a field k of characteristic m p, with x P K and GFppmś q Ă K, then by taking a root y of Y p ´ Y ` x m we have Y p ´ Y ` x “ iPGFppm q rY ´ py ` iqs and hence exactly as in ˘ ` m the p-cyclic case we get Gal Y p ´ Y ` x, K “ pz p qm . By throwing away a root of Fr1 n,t,s,a of degree n ´ 1 in Y with coefficients in kpXq given by “ ‰ Fr1 n,t,s,a “ Y ´1 rpY ` 1qn ´ 1s ´ aX ´s Y ´1 pY ` 1qt ´ 1 ´ ¯ 1 1 r r and for its Galois group G n,t,s,a “ Gal F n,t,s,a , kpXq , in my joint paper [15] with Popp and Seiler it is shown that: (7.11 ) 1 “ t “ n ´ 11 and p “ 3 and S ” 0pn ´ 1q ñ Gr1 n,t,s,a “ PS Lp2, 11q, where, for the said values of the parameters, the equation Gr1 n,t,s,a “ 0 gives an unramified covering of Lk . Finally let pdq

F n,q,s,a,b “ Y dn ´ aX x Y dt ` b with positive integer d ı 0ppq where once again a, b are nonzero elements of k and n, t, s are positive integers with t ă n and GCDpn, tq “ 1 and n ´ t “ q “ a positive pdq

pdq

power of p. For the Galois group Gn,q,s,a,b “ GalpF n,q,s,a,b , kpXqq, in my joint paper [15] with Popp and Seiler it is shown that: pdq

(7.21 ) d “ t “ 2 “ n ´ 9 and q “ 9 and p “ 3 ñ Gn,q,s,a,b “ ˚ « M , where, for the said values of the parameters, the equation M11 11 pdq

Gn,q,s,a,b “ 0 gives an unramified covering of Lk . ˚ and M are isomorphic as abstract groups but 18 Again note that M11 11 ˚ is the transitive but not 2-transinot asa permutations groups; here M11 tive incarnation of M11 obtained by considering its cosets according to the index 22 subgroup PS Lp2, 9q. 21

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8 Grothendieck Having dropped my prejudice against Statistics, it is high time to show my appreciation of Grothendieck. For example by using the (very existential and highly nonequational) work of Grothendieck [33] on tame coverings of curves, in [7] and [11] I have shown that for any pairswise nonisomorphic nonabelian finite simple groups D1 , . . . , Du with | Aut Du | ı 0ppq the wreath product pD1 ˆ ¨ ¨ ¨ ˆ Du q Wr Z p belongs to πA pLk q. For instance we may take D1 “ Am1 , . . . Du “ Amu with 4 ă M1 ă M2 ă . . . ă Mu ă p. Actually, using Grothendieck [33] I first prove an Enlargement Theorem and then from i deduce the above result about wreath products as a group theoretic consequence. The Enlargements Theorem asserts that if Θ is any πA pLk q, then some enlargement of Θ by J belongs to πA pLk q. Now enlargement is a generalization of group extensions. Namely, an enlargement of an group Θ by a group J is group G together with an exact sequence 1 Ñ H Ñ J Ñ 1 and a normal subgroup ∆ of λpHq, where λ is the given map of H into G, such that λpHq{∆ is isomorphic to Θ and no nonidentity normal subgroup of G is contained in ∆. Note that here G is an extension of H by J. The motivation behind enlargements is the fact that a Galois extensions of A Galois extension need not be Galois and if we pass to the relevant least Galois extension then its Galois group is an enlargement of the second Galois group by the first. Talking of group extensions, as a striking consequence of CT it can be seen that the direct product of two finite nonabelian simple groups is the only extensions of one by the other. Here the relevant direct consequence of CT is the Schreier Conjecture which which says that the outer automorphism group of any finite nonabelian simple group is solvable; see Abhyankar [11] and Gorenstein [32]. As another interesting result, in [12] I proved that πA pLk q is closed with respect to direct products. it should also be noted that Nori [49] has shown that πA pLk q contains SLpn, pm q and some other Lie type simple groups of characteristic p. Returning to Grotherndieckian techniques, Serre [56] proved that if πA pLk q contains a group H then it contains every quasi p-group which 22

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23

is an extension of H by a solvable group J. 19 Indeed it appears that the ongoing work of Harbarter and Raynaud using Grothendieckian techniques is likely to produce existential proofs of the quasi p-group conjecture. But it seems worthwhile to march on with the equational concrete approach at least because it gives results over the prime field GFppq and also because we still have no idea what tha complete algebraic fundamental group πCA pLk q looks like where πCA pLk q is the Galois group over kpXq of the compositum of all finite Galois extensions of kpXq which are ramified only at infinity and which are contained in a fixed algebraic closure of kpXq.

9 Ramanujan In the equational approach . “modula” things seem destined to plays a significant role. For instance the Carlitz-Serre construction PSLp2, qq coverings and Serre’s alternative proof [57] that Gn,q,s,q “ PSLp2, 8q for q “ p “ 7 and n “ 9, are both modular. Similarly my joint paper [16] with Popp and Seiler which uses the Klein and Macbeath curves for writing down PSLp2, 7q and PSLp2, 8q coverings for small characteristic is also modular in nature. Inspired by all this, I am undertaking the project of browsing in the 2 volume treatise of Klein and Fricke [43] on Elliptic Modular Functions to prepare mysely for understanding Ramanujan himself who may be called the king of Things Modular, where Things = Funcitons, equations, Mode of Thought or what have you; see Ramanujan’s Collected Papers [50] and Ramanujan Revisited [15]. To explain what are moduli varieties and modular funtions in a very naive but friendly manner: The discriminant b2 ´ 4ac of a quadratic aY 2 `bY `c is the oldest known invariant. Coming to cubics or quartics aY a `bY 3 `cY 2 `e we can, as in books on theory of equations, consider algebraic invariants, i.e., polynomial functional of a, b, c, d, e which do not change (much) when we change Y by a fractional linear transformation (see my Invariant Theory Paper [13]), or we may consider transcen23

24

20

Shreeram S. Abhyankar

denatal invariants and then essentially we ge ellipatic modular functions. More generally we may consider several (homogeneous) polynomials in several (set-of) variables; when thought of as funcitions of the variables they give us algebraic varieties or multi-periodic functions or abelian varieties and so on; but as functions of the coeficients we get algebraic invariants or moduli varieties or modular functions. Modular functions and their transforms are related by modular equations; thinks of the expansion of sin nθ in terms in sin θ! But postponing this to another lecture on another day, let me end with a few equational questions suggested by the experimental data presented in this lecture. Question 9.1. Which quasi p-groups can be obtained by coverings of a line by a line? In other words, which quasi p-groups are the Galois groups of f over kpXq for some monic polynomial f in Y with coefficients in krXs such that f is linear in X, no valuation of kpXq{k is ramified in the splitting field of f other than the valuations X “ 0 and X “ 8, and the may even allow the given quasi p-group to equal ppGalpF, kpXqqq ; note that Galp f, kpXqq{ppGalp f, kpXqqq is necessarily a cyclic group of order prime to p. In any case this prime to p cyclic quotient as well as the tame branch point at X ´ 0 can be removed be Abhyankar’s Lemma. [Hoped for Answer: many quasi p-groups if not all]. Question 9.2. Do fewnomilas suffice for all simple quasi p-groups? Etymology: binomial, trinomial, . . . , fewnomial. In other words,is there a positive integer d (hopefully small) such that every simple quasi p-group can be realized as the Galois group of a polynomial f containing at most d terms in Y (more precisely, at most d monomials i Y) with coefficients in krXs which gives an unramified covering of the X-axis Lk ? Indeed,do fewnomias suffice for most (if not all) quasi p-groups (without requiring them to be simpel)? If not, then do sparanomials suffice for most (if not all) quasi p-groups? Etymology: sparnomial = spare polynomial in Y plus a polynomial in Y p .

24

Fundamental Group of the Affine Line in Positive Characteristic

25

Question 9.3. Which quasi p-groups can be realized as Galois groups of polynomials in Y whose coeffcients are polynomials in X over the prime field GFppq such that no valuation of GFppqrXs is ramified in the relevant splitting field and such tha GFppq is relatively algebraically closed in the splitting field and such that GFppq is relatively. algebraiclally closed in that splitting field? Same question where we drop the condition of GFppq begin relatively algebraically closed but where we repalce quasi p-groups by finite groups G for which G{ppGq is cyclic. For instance, given any positive power q1 of any prime p1 such that the order of PSLp2, q1 q is divisible by p, we may ask whether there exists an unramified covering of the affine line over GFppq whose Galois group is the semidirect product of PS Lp2, q1 q with AutpGFpq1 qq, i.e., equivalently, whether there exists a polynomial in Y over GFppqrXs, with Galois group the said semidirect product, such that no valuation of GFppqrXs is ramified in the relevant splitting field (without requiring GFppq to be relatively algebraically closed in that splitting field). Note that, in [9], this last question has been answered affirmatively for p “ 7 and q1 “ 8. Also note that for even q1 the said semidirect product is the projective 21 semilinear group PΓLp2, q1 q, whereas for odd q1 it is an index 2 subgroup of PΓLp2, q1 q; for definitions see [8]. Question 9.4. Do we get fewer Galois groups if we replace branch locus by discriminant locus? For instance, can every quasi p-group be realized as the Galois group of a monic polynomial in Y over krXs whose Y-discriminant is a nonzero element of k? Likewise which members of πA pLk,w q, where Lk,w “ Lk minus w points, can be realized as Galois groups of monic polynomials in Y over krXs whose Y-discriminants have no roots other than the assigned w points. We may ask the same thing also for ground fields of characteristic zero. Question 9.5. Concerining the bar and tilde equations discussed in (6.1) to (6.8) and (7.1*) to (7.5*) respectively, what further interesting groups do we out of the border values of t?

25

26

Shreeram S. Abhyankar

Question 9.6. Can we describe the complete algebraic fundamental group πCA pLk q? More generally, for a nonsigular projective curve Cg of genus g minus w ` 1 points, can we describe the complete algebraic fundamental πCA pCg,w q? Question 9.7. Descriptively speaking, can the “same” equation give unramified coverings of the affine line for all quasi p-groups in the “same family” of groups? For instance, Y q`1 ´ XY ` 1 “ 0 gives an unramified covering of the affine line, over a field of characteristic p, with Galois group PS Lp2, qq for every power q every prime p. Now thinking of the larger family of groups PS Lpm, qq, can be find a ‘single” equation with integer coefficients, “depending” on the parameters m and q, giving an unramified covering of the affine line, over a field of characteristic p, whose Galois group is PS Lpm, qq for every integer m ą 1 and every power q of every prime p? Can we also arrange that the “same” equation gives an unramified covering of the affine line, over every field whose characteristic divides the order of PSLpm, qq, whose Galois group is PSLpm, qq? Even more, can we arrange that at the same time the Galois group of that equation over QpXq is PSLpm, qq (but no condition on ramification) where Q is the algebraic closure of Q? Note 9.8. Two or more of the above questions can be combined in an obvious manner to formulate more questions.

References 22

[1] S.S. Abhyankar, On the ramification of algebraic functions, Amer. Jour. Math. 77 (1955) 572–592. [2] S.S. Abhyankar, Local uniformization on algebraic surfaces over ground fields of characteristics p ‰ 0, Ann. Math. 63 (1956) 491– 526. [3] S.S. Abhyankar, Coverings of algebraic curves, Amer. Jour. Math. 79 (1957) 825–856. 26

Fundamental Group of the Affine Line in Positive Characteristic

27

[4] S.S. Abhyankar, Tame coverings and fundamental groups of algebraic varieties, Parts I to VI, Amer.Jour.Math. 81, 82 (1959–60). [5] S.S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976) 409–448. [6] S.S. Abhyankar, Algebraic Geometry for Scientists and Engineers, Mathematical Surveys and Monographs 35, Amer. Math. Soc., Providence, 1990. [7] S.S. Abhyankar, Group enlargements, C. R. Acad. Sci. Paris 312 (1991) 763–768. [8] S.S. Abhyankar, Galois theory on the line in nonzero characteristic, Bull. Amer. Math. Soc. 27 (1992) 68–133. [9] S.S. Abhyankar, Square-root parametrization of plane curves, Algebraic Geometry and Its Applications, Springer-Verlag (1994) 19–84. [10] S.S. Abhyankar, Alternating group coverings of the affine line in characteristic greater than two, Math. Annalen 296 (1993) 63–68. [11] S.S. Abhyankar, Wreath products and enlargements of groups, Discrete Math. 120 (1993) 1–12. [12] S.S. Abhyankar, Linear disjointness of polynomials, Proc. Amer. Math. Soc. 116 (1992) 7–12. [13] S.S. Abhyankar, Invariant theory and enumerative combinatories of Young tableaux, Geometric Invariance in computer Vision, Edited by J. L. Mundy and A. Zisserman, MIT Press (1992) 45–76. [14] S.S. Abhyankar, J. Ou and A. Sathaye, Alternating group coverings of the affine line in characteristic two, Discrete Mathematics, (To Appear). [15] S.S. Abhyankar, H. Popp and W. K. Seiler, Mathieu group cover- 23 ings of the affine line, Duke Math. Jour. 68 (1992) 301–311. 27

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Shreeram S. Abhyankar

[16] S.S. Abhyankar, H. Popp and W. K. Seiler, Construction techniques for Galois coverings of the affine line, Proceedings of the Indian Academy of Sciences 103 (1993) 103–126. [17] S.S. Abhyankar and I. Yie, Small degree coverings of the affine line in characteristic two, Discrete Mathematics (To Appear). [18] C. Arf, Untersuchungen iiber quadratische Formen in K¨orpern der Characteristik 2(Teil I.), Crelle Journal 183 (1941) 148–167. [19] E. Berlekamp, Algebraic Codign Theory McGraw-Hill, 1968. [20] E. Berlekamp, An analog to the discriminant over fields of characteristic 2, Journal of Algebra 38 (1976) 315–317. [21] R. C. Bose, On the construction of balanced incomplete block designs, Ann, Eugenics 9 (1939) 353–399. [22] P.J. Cameron, Finite permutation groups and finite simple groups, Bull. Lond. Math. Soc. 13 (1981) 1–22. [23] P.J. Cameron. Permutation groups, Handbook of Combinatorics, Elsevier, Forthcoming. [24] P.J. Cameron and J. Cannon, Fast recongnition of doubly transitive groups, Journal of Symbolic Computations 12 (1991) 459–474. [25] L. Carlitz, Resolvents of certain linear groups in a finite field, Canadian Jour. Math.8 (1956) 568–579. [26] C. Chevalley, Theory of lie Groups, Princeton University Press, 1946. [27] G. Chrystal, Textbook of Algebra, Parts I and II, Edinburgh, 1886. [28] R, Dedekind and H. Weber, Theorie der Algebraischen Funktionen einer Ver¨anderlichen, Crelle Joournal 92 (1882) 181–290. [29] P. Dembowski, Finte Geometries, Spirnger-Verlag, 1968. 28

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[30] W, Feit On a class of doubly transitive permutation groups, Illinois Jour. Math. 4 (1960) 170–186. [31] A. R. Forsyth, Theory of Functions of a Complex Variable, Cambridge University Press, 1918. [32] D. Gorenstein, Classifying the finite simple groups, Bull. Amer. 24 Math. Soc. 14 (1986) 1–98. ´ [33] A. Grothendieck, Revˆetements Etales et Groupe Fondamental (SGA 1), Lecture Notes in Mathematics 224 Springer-Verlag, 1971. [34] D.G. Higman, Finite permutation groups of rank 3, Math. Z. 86 145–156. ¨ [35] D. Hilbert, Uber das Dirichletesche Prinzip, Mathematische Annalen 59 (1904) 161–186. [36] B. Huppert and N. Blackburn, Finite Groups I, II, III, SpringerVerlag, New York, 1982. [37] N. Jacobson, Lectures in Abstract Algebra, Vol III, Van Nostrand, Princeton, 1964. [38] N. Jacobson, Basic Algebra, Vol I, W. H. Freeman and Co. San Francisco, 1974. [39] C. Jordan, Sur la limite de transitivit´e des groupes non altern´es, Bull. Soc. Math. France 1 (1973) 40–71. [40] H. W. E. Jung,Darstellung der Funktionen eines algebraischen K¨orperszweier unabh¨angigen Ver¨angigen Ver¨anderlichen in der Umgebung einer Stelle, Crelle Journal 133 (1908) 289–314. [41] W. M. Kantor, Homogeneous designs and geometric lattices, Journal of Combinatorial Theory, Series A 38 (1985) 66–74. 29

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[42] W. M. Kantor and R. L. Liebler, The rank 3 permutation representations of the finite classical groups, Trans. Amer. Math. Soc. 271 (1982) 1–71. [43] F. Klein and R. Fricke, Vorlesungen u¨ ber die Theorie der Elliptischen Modulfunctionen, 2 volumes, Teubner, Leipzig, 1890. [44] M. W. Liebeck, The affine permutation groups of rank three, Proc. Lond. Math. Soc. 54 (1987) 447–516. ¨ [45] B. Marggraff, Uber primitive Gruppern mit transitiven Untergruppen geringeren Grades, Dissertation, Giessen, 1892. [46] E. Mathieu, M´emoire sur l’´etude des fonctions de plusieurs quantit´es, sur la mani`ere de les former, et sur les substitutions qui les laissent invariables, J. Math Pures Appl. 18 (1861) 241–323. 25

[47] I. Newton, Geometria Analytica 1680. ¨ [48] M. Noether, Uber einen Satz aus der Theorie der algebraischen Funktionen, Math. Annalen 6 (1873) 351–359. [49] M .V. Nori, Unramified coverings of the affine line in positive characteristic Algebraic Geometry and Its Applications, SpringerVerlag (1994) 209–212. [50] S. Ramanujan, Collected Papers, Cambridge University Press, 1927. [51] S. Ramanujan, Ramanujan Revisited, Proceedings of the Centenary Conference at University of Illinois, Edited by G. E. Andrews, R. A. Askey, B. C. Berndt, K. G. Ramanathan and R. A. Rankin, Academic Press, 1987. [52] B. Riemann, Grundlagen f¨ur eine allgemeine Theorie der Functionen einer ver¨anderlichen complexen Gr¨osse, Inauguraldissertation, G¨ottingen (1851). Collected Works, Dover (1953) 3–43. 30

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31

[53] B. Riemann, Theorie der Abelschen Funktionen, Jour. f¨ur die reine undang. Mathematik 64 (1865) 115–155. [54] H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, 1934. [55] J-P. Serre, G´eom´etrie alg´ebrique et g´eom´etrie analytique, Ann. Inst. Fourier 6 (1956) 1–42. [56] J-P. Serre, Construction de revˆetments e´ tales de la droite affine en charact´eristic p, C. R. Acad. Sci. Paris 311 (1990) 341–346. [57] J-P. Serre, A letter as an appendix to the squar-root parametrization paper of Abhyankar, Algebraic Geometry and Its Applications, Springer-Verlag (1994) 85–88. [58] S. S. Shrikhande, The impossibility of certain symmetrical balanced incomplete block designs, Ann. Math. Statist. 21 (1950) 106–111. [59] M. Suzuki. On a class of doubly transitive groups, Ann. Math. 75 (1962) 104–145. [60] H. Wielandt. Finite Permutation Groups, Academic Press, New York, 1964. [61] E. Witt. Theorie der quadratischen Formen in beliebigen K¨orpern, Crelle Journal 176 (1937) 31–41. [62] H. Zassenhaus, Kennzeichung endlicher linearen Gruppen als Pre- 26 mutationgruppen, Abh. Mth. Sem. Univ. Hamburg (11) (1936) 17– 744.

Mathematics Department Purade University, West Lafayette, IN 47907 U.S.A. 31

Impact of geometry of the boundary on the positive solutions of a semilinear Neumann problem with Critical nonlinearity Adimurthi Dedicated to M.S. Narasimhan and C.S. Seshadri on their 60th Brithdays 27

Let n ě 3 and Pn be a bounded domain with smooth boundary. We are concerned with the problem of existence of a function u satisfying the nonlinear equation ´∆u “ u p ´ λu uą0 Bu “ 0 on Bv

in Ω

BΩ

(1)

1{pp´1q is a solution (1) and we where p “ n`2 n´2 , λ ą 0. Clearly u “ λ call it a trivial solution. The exponent p “ n`2 n´2 is critical from the view point of Soblev imbedding. Indeed the solution of (1) corresponds to critical points of the functional ş ş 2 2 Ω |∇u | dx ` λ Ω u dx (2) Qλ puq “ `ş ˘ p`1 dx 2{p`1 |u| Ω

on the manifold

* " ż M “ u P H 1 pΩq; |u| p`1 dx “ 1

(3)

Ω

In fact, if v ď 0 is a critical point of (2) on M, then u ´ Qpvq1{pp´1qv 2n satisfies (1). Note that p ` 1 “ n´2 is the limiting exponent for the 1 2n{pn´2q imbedding H pΩq ÞÑ L pΩq. Since this imbedding is not compact, the manifold M is not weakly closed and hence Qλ need not satisfy 32

Impact of geometry of the boundary on the positive solutions...

33

the Palais Smale condition at all levels. Therefore there are serious difficulties when trying to find critical points by the standard variational methods. In fact there is a sharp contrast between the sub critical case 28 n`2 p ă n`2 n´2 and the critical case p “ n´2 . Our motivation for investigation comes from a question of Brezis [11]. If we replace the Neumann condition by Dirichlet condition u “ 0 on BΩ in (1), then the existence and non existence of solutions depends in topology and geometry of the domain (see Brezis-Nirenbreg [14], Bhari-Coron [9], Brezis [10]). In view of this, Brezis raised the following problem “Under what conditions on λ and Ω, (1) admits a solution?” The interest in this problem not only comes from a purely mathematical question, but it has application in mathematical biology, population dynamics (see [16]) and geometry. In order to answer the above question let us first look at the subcritical case where the compactness in assured. Subcritical case 1 ă p ă

n`2 n´2 .

This had been studied extensively in the recent past by Ni [18], and Lin-Ni-Takagi [16]. In [16], Lin-Li-Takagi have proved the following Theorem 1. There exist two positive constants λ˚ and λ˚ such a) If λ ă λ˚ , then (1) admits only trivila solutions. b) If λ ą λ˚ , then (1) admits non constant solutions. Further Ni [18] and Lin-Ni [15] studied the radial case for all 1 ă p ă 8 and proved the following Theorem 2. Let Ω “ x; |x| ă 1 is a ball. Then there exists two positive constants λ˚ and λ˚ such that λ˚ ď λ˚ such that λ˚ ď λ˚ and a) For 1 ă p ă 8, λ ą λ˚ , (1) admits a radially increasing solution b) if p ‰ n`2 n´2 , then for 0 ă λ ă λ˚ , (1) does not admit a non constant radial solution. 33

34

Adimurthi c) Let Ω “ tx; 0 ă α ă |x| ă βu be an annuluar domain and 1 ă p ă 8. Then there exist two positive constants λ˚ ď λ˚ such that for λ˚ ě λ˚ , (1) admits a non constant radial solution and if λ ď λ˚ , then (1) does not admit a non constant radial solution. In view of these results Lin and Ni [15] made the following

29

Conjecture. Let p ě such that

n`2 n´2

then there exist two positive constants λ ď λ˚

(A) For 0 ă λ ă λ˚ , (1) does not admit non constant solutions. (B) For λ ă λ˚ , (1) admits a non constant solution. In this article we analyze this conjecture in the critical case p “ Surprisingly enougn, the critical case is totally different from the subcritical. In fact the part (A) fo the conjecture in general is false. The following results of Adimurthi and Yadava [4] and Budd, Knaap and Peletier [12] gives a counter example to the Part (A) of the conjecture. n`2 n´2 .

Theorem 3. Let n “ 4, 5, 6 and Ω “ tX : |x| ă 1u. Then there exist a λ˚ ą 0 such that for 0 ă λ ă λ˚ , (1) admits a radially decreasing solution. Let us now turn our attention to part (B) of the conjecture. Let S denote the best Sobolev constant for the imbedding H 1 pRn q ÞÑ L2n{pn´2q pRn q given by * "ż ż 2n{n´2 2 |u| dx “ 1 (4) S “ inf |∇u| dx : Rn

Rn

Then S is achieved and any minimizer in given by Uε,x0 for some ε ą 0, x0 P Rn where „

n´2 2 npn ´ 2q Upxq “ npn ´ 2q ` |x|2 ˙ ˆ x ´ x0 1 Uε,x0 pxq “ n´2 U ε ε 2 34

(5) (6)

Impact of geometry of the boundary on the positive solutions...

35

In order to answer part (B) of the conjecture, geometry of the boundary play an important role. To see this, we look at a more general problem than (1), the mixed problem. Let BΩ “ Γ0 Y Γ1 , Γ0 X Γ1 “ φ, Γi are submanifolds of dimension pn ´ 1q. The problem is to find function u satisfying n`2

´∆u ` λu “ u n´2

in

Ω

uą0

u “ 0 on Γ0 Bu “ 0 on Γ1 Bv

Let

H 1 pΓ0 q “ u P H 1 pΩq : u “ 0 on Γ0 ( S pλ, Γ0 q “ inf Qλpuq ; u P H 1 pΓ0 q X M

(7)

(

(8) (9)

Clearly, if S pλ, Γ0 q is achieved by some v, then we can take v ď 0 n´2 and u “ S pλ, Γ0 q 4 v satisfies (7). u is called a minimal energy solution. Existence of a minimal energy solution is proved in Adimurthi and Mancini [1] (See also X.J. Wang [22]) and have the following Theorem 4. Assume that there exist an x0 belonging to the interior of Γ1 such that the mean curvature Hpx0 q at x0 with respect to unit outward normal is positive. Then S pλ, Γ0 q is achieved. Sketch of the Proof. The proof consists of two steps. Step 1. Suppose S pλ, Γ0 q ă S {22{n , then S pλ, mΓ0 q is achieved.

Let vk P H 1 pΓ0 q X M be a minimizing sequence. Clearly tvk u is bounded in H 1 pΩq. Let for subsequence of tvk u still denoted by tvk u, coverges weakly to v0 and almost everywhere in Ω. We first claim that v0 ı 0. Suppose v0 ” 0, then by Cherrier imbedding (See [8]) for every ε ą 0, there exists Cpεq ą 0 such that ˆż ˙2{p`1 p`1 1“ |vk | dx Ω

35

30

36

Adimurthi 22{n p1 ` εq ď S

ż

2

Ω

|∆vk | dx ` Cpεq

ż

v2k dx.

By Rellich’s compactness, vk Ñ 0 in L2 pΩq and hence in the above inequality letting k Ñ 8 and ε Ñ 0 we obtain 22{n p1 ` εq lim Qλ pvk q εÑ0 S kÑ8 2{n 2 S pλ, Γ0 q “ S ă1

1 ď lim

which is a contradiction. Hence v0 ı 0. Let hk “ vk ´ V0 , then hk Ñ 0 weakly in H 1 pΩq and strongly in L2 pΩq. Hence S pλ, Γ0 q “ Qλ pvk q ` 0p1q ˆż ˙ s{p`1 ż p`1 “ Qλ pv0 q |v0 | ` |∆hk |2 dx ` 0p1q Ω

Ω

Now by Brezis-Lieb Lemma, Cherrier imbedding, from the above inequality, and by the hypothesis, we have for sufficiently small ε ą 0, ˆż ˙2{p`1 p`1 S pλ, Γ0 q “ S pλ, Γ0 q |vk | dx Ω #ˆż + ˙ ˆż ˙ 2{p`1

ď S pλ, Γ0 q ` 0p1q

“ S pλ, Γ0 q

“ S pλ, Γ0 q

Ω

#ˆż

Ω

ˆż

Ω

|v0 | p`1 dx

|v0 | ż

p`1

Ω

|v0 |

dx

2{p`1

`

˙2{p`1 2

*

`

Ω

|hk | p`1 dx

22{n p1 ` εqˆ S

|∇hk | dx ` 0p1q

p`1

dx

˙2{p`1 36

`

ż

Ω

|∇hk |2 dx ` 0p1q

Impact of geometry of the boundary on the positive solutions... “ S pλ, Γ0 q

ˆż

Ω

|v0 |

ˆż

Ω

p`1

|v0 |

dx

˙2{p`1

p`1

dx

37

` S pλ, Γ0 q ´ Qλ pv0 qˆ

˙2{p`1

this implies that Qλ pv0 q ď S pλ, Γ0 q. Hence v0 is a minimizer. Step 2. S pλ, Γ0 q ă S {22{n

Let x0 belong to the interior of Γ1 at which Hpx0 q ą 0 and r ą 0 that Bpx0 , rq X Γ0 “ φ. Let ϕ P C08 pBpx0 , rqq such that ϕ “ 1 for |x ´ x0 | ă r{2. Let ε ą 0 and vε “ ϕUε,x0 . Then vε P H 1 pΓ0 q and we can find positive constants An and an depending only on n such that Qλpvε q “ where

S ´ An Hpx0 qβ1 pεq ` an λβ2 pεq ` 0pβ1 pεq ` β2 pεqq 22{n (10) # ε log1{ε if n “ 3 β1 pεq “ ε it n ě 4 $ ’ it n ě 4 &ε 2 β2 pεq “ ε log 1{ε if n “ 4 ’ % 2 ε it n ě 4

Hence for ε small and since Hpx0 q ą 0 we obtain Qλ pvε q ă S {22{n and this proves Step (31) and hence the theorem. Now it is to be noted the the curvature condition on Γ1 is very essential. If the curvature conditions fails, then in general (7) may not admit any solution. Example. Let B “ x : |x| ă 1 and

Ω “ x P B : xn ą 0 37

31

38

Adimurthi Γ1 “ x P BΩ : xn “ 0

Γ0 “ x P BΩ : xn ą 0

Let u P H 1 pΓ0 q be a solution of (7). Define w on B by # wpx1 , xn q if xn ą 0 wpx1 , xn q “ wpx1 , xn q if ´xn ă 0 32

Since

Bu Bv

“ 0 on Γ1 , w satisfies n`2

´∆w ` λw “ w n´2 in B wą0

w “ 0 on BB. Hence by Pohozaev’s identity we obtain ´λ

ż

2

B

w dx “

ż

BB

|∇w|2 xx, vydξ

Hence by a contradiction. Notice that the mean curvature is zero on Γ1 .

Proof of Part (B) of the Conjecture Let Γ1 “ BΩ. Since BΩ is smooth, we can find an x0 P BΩ such that Hpx0 q ą 0. Hence from theorem (4), (1) admits a minimal energy S ˚ solution uλ . Let u0 “ λ1{p´1 and λ˚ “ p2|ω|q 2{n . Then λ ą λ Qλ puλ q ă

S ă λ|Ω|2{n “ Qλ pu0 q 22{n

Hence uλ is a non constant solution of (1) and this proves part (B) of the conjecture. 38

Impact of geometry of the boundary on the positive solutions...

39

Properties of the minimal energy solutions 1. By Theorem 3 part (A) of conjecture in general is flase. Now we can ask whether this is true among minimal energy solution? In fact it is true. The following is proved in Adimurthi-Yadava [6]. Theorem 5. There exist a λ˚ ą 0 such that for all 0 ă λ ă λ˚ , the minimal energy solution are constant. Proof. By using the blow up techinque [13], we can prove that for every ε ą 0 there exists a aλpεq ą 0 such that for 0 ă λ ă λpεq, if uλ is a minimal energy solution, then |uλ |8 ď ε

(11)

where | ¨ |8 denotes the L8 norm. Let µ1 be the first non zero eigenvalue of ´∆ψ “ µψ in Ω Bψ “ 0 on BΩ. Bv

Let uλ “

1 ş uλ dx and ϕλ “ uλ ´ uλ . Then ϕλ satisfies |Ω| Ω ´∆ϕλ ` λϕλ “

p uλ

`p

ż1 0

puλ ` tϕλ q p´1 ϕ2λ dtdx

ş From (11) we have 0 ď uλ ` tϕλ ď uλ ď ε and Ω ϕλ dx “ 0. Therefore we obtain ż ż ż ˘ ` p´1 2 2 2 ϕ2λ dx |∇ϕλ | ` λϕλ dx ď pε pµ1 ` λq ϕλ dx ď Ω

Ω

Ω

µ1 and λ˚ “ λpεq, then the above inequality Now choose ε p´1 “ 2p implies that ϕλ ” 0 and hence uλ is a constant. This proves the theorem.

39

33

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Adimurthi

2. Concentration and multiplicity results. From the concentration compactness results of P.L. Lions [17] if uλ is a minimal energy solution of (1), then for anu sequence λk Ñ 8 with |∇uλk |2 dx Ñ dµ, n{2 there exist a x0 P BΩ such that dµ “ s 2 δ x0 . Now the natural question in “is it possible to characterize the concentration points x0 ”? One expects from the asymptotic formula (10) that x0 must be a point of maximum mean curvature. This has been proved in Adimurthi, Pacella and Yadava [7] and we have the following Theorem 6. Let uλ be a minimal energy solution of (1) and pλ P Ω be such that ! ) uλ pPλ q “ max uλ pxq; x P Ω then there exist aλ0 ą 0 such that for all λ ą λ0 a) pλ P BΩ and is unique, b) Let n ě 7. The limit points of tPλ u are contained in the points of maximum mean curvature. Part (a) of this Theorem is also proved in [19]. In view of the concentration at the boundary, it follows that the minimal energy solutions are not radial for λ sufficiently large and Ω beging tha ball. Hence in a ball, for large λ, we obtain at least two solutions one radial and the other non radial (see [5]). If Ω is not a ball then in Adimurthi and Mancini [2], they obtained that CatBΩ pBΩ` q number of solutions for (1) where BΩ` is the set of points in BΩ where the mean curvature is positive (here for X Ă U, Y topological space, then CatY pxq is category of X in Y). Further if BΩ has rich geometry in the sense described below, then Adimurthi-Pacella and Yadava [7] have obtained more solutions of (1). They have proved the following 34

Theorem 7. Let n ě 7. Assume that BΩ has k-peaks, that is there exist k-points x1 , . . . xk Ă BΩ at which Hpxi q is strictly local maxima. Then there exists aλ0 ą 0 such that for λ ą λ0 , there are k distinct solutions tuiλ uki “ 1 of (1) such that uiλ concentrates at xi as λ Ñ 8. 40

Impact of geometry of the boundary on the positive solutions...

41

Theorems 6 and 7 has been extended for the mixed boundary value problems. Theorem 7 is not applicable in the case when Ω is a ball. On the other hand, given a positive integer k, there exists a λpkq such that for λ ą λpkq, (1) admits at least k number of radial solutions (see [18]). Part (B) of the conjecture gives infinitely many rotationally equivalent solutions of minimal energy. In view of this it is not clear how to obtain more non radial solutions which are not rotationally equivalent.

References [1] Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical non-linearity, A tribute is honour of G. Prodi, Ed by Ambrosetti and Marino, Scuola Norm. Sup. Pisa (1991) 9-25. [2] Adimurthi and G, Mancini, Effect of geometry and topology of the boundary in the critical Neumann problem, R. Jeine Angew. Math, to appear. [3] Adimurthi and S.L. Yadava, Critical Sobolev exponent problem in Rn pn ě 4q with Neumann boundary condition, Proc. Ind. Acad. Sci. 100 (1990) 275-284. [4] Adimurthi and S.L. Yadava, Existence and nonexistence of positive radila solutions of Neumann problems with Critical Sobolev exponents, Arch. Rat. Mech. Anal. frmros´´00 (1991) 275-296. [5] Adimurthi and S.L. Yadava, Existence of nonradial positive solution for critical exponent problem withb Neumann boundary condition, J. Diff. Equations, 104 298-306. [6] Adimurthi and S. L. Yadava, On a conjecture of Lin-Ni for semilinear Neumanna problem, Trans. Amer. Math. Soc. 336 (1993) 631-637. 41

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[7] Adimurthi, F. Pacella and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Newmann problem with Critical non linearity, J. Funct. Anal. 113 (1993) 318-350. 35

[8] T. Aubin, Nonlinear analysis of manifold: Monge-Ampere equations, New York, Springer-Verlag (1992). [9] A. Bhari and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988) 253-290. [10] H. Brezis, Elliptic equations with limiting Sobolev exponent. The impact of topology, Comm. Pure appl. Math. 39 (1996) S17-S39. [11] H . Brezis, Non linear elliptic equations involving the Critical Sobolev exponent in survey and prespectives, Directions in Partical differential equations, Ed, by Crandall etc. (1987), 17-36. [12] C. Budd, M.C. Knapp and L.A. Peletier, Asymptotic behaviour of solution of elliptic equations with critical exponent and Neumann boundary conditions, Proc, Roy. Soc. Edinburg 117A (1991) 225250. [13] B. Gidas and J. Spruck, A priori boundas for positivce solutions of nonlinear elliptic equations, Comm. Part. Diff. Equations 6 (1981) 883-901. [14] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving Critical exponents, Comm. Pure Appl. Maths. Vol.36 (1983) 437-477. [15] C.S. Lin and W. M. Ni, On the diffusion coefficient of a semilinear Neumann problem, Springer Lecture Notes 1340 (1986). [16] C.S. Lin and W. M. Ni and I. Takagi, Large amplitude stationary solutions to an chemotaxis system, J. Diff. Equations 72 (1988) 1-27. 42

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[17] P.L. Lions, The concentration compactness principles in the calcular of variations: The limit case (Part 1 and part 2), Riv. Mat. Iberoamericana 1 (1985) 145-201; 45-121. [18] W. M. Ni, On the positive radial solutions of some semi-linear elliptic equations on Rn , Appl. Math. Optim. 9 (1983) 373-380. [19] W. M. Ni, S. B. Pan and I. Takagi. Singular behaviour of least energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents, Duke. Math. 67 (1992) 1-20. [20] W.M. Ni and I . Takagi, On the existence and the shape of solutions to an semi-linear Neumann problem, to appear in Proceedings of the conference on Nonlinear diffusion equations and their equilibrium states; held at Gregynog, Wales, August 1989. [21] W.M. Ni and I. Takagi, On the shape of least-energy solutions to a 36 semilinear Neumann problem, Comm. Pure Appl. Math. 45 (1991) 819-851. [22] X. J. Wang, Neumann problem of semi-linear elliptic equations involving critical Sobolev exponents, J. Diff. Equations, 93 (1991) 283-310.

T.I.F.R. Centre P.O. Box No. 1234 Bangalore 560 012.

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Sur la cohomologie de certains espaces de ´ vectoriels modules de fibres Arnanud Beauville* D´edi´e a` M.S. Narasimhan et C.S. Seshadri pour leur 60e`me anniversaire 37

Soit X une surface de Riemann compacte. Fixons des entiers r et d premiersm entre eux, avec r ě 1, et notons M 1’ espace des modules U x pr, dq des fibri´es d. C’est une vari´et´e projective et lisse, et il existe un fibr´e de Poinca´e E sure X ˆ M; cela signifie que pour tout point e de M, correspondant a` un fibr´e E sur X, la restriction de E a` X ˆ e est isomorphe a` E. Notons p, q les projections de X ˆ M sur X et M respectivement. Soit m un entier ď r; la classe de Chern cm pEq admet une d´ecomposition de K¨unneth ÿ cm pEq “ p˚ ξi ¨ q˚ µi , i

H ˚ pX, Zq, µi

H ˚ pM, Zq, degpξ

avec ξi P P i q ` degpµi q “ 2m. Nous dirons que les classes µi sont les composantes de K¨unneth de cm E. Un des re´sultats essientiels de [A-B] est la d´etermination d’un ensemble de g´en´erateurs de l’alg´ebre de cohomologie H ˚ pM.Zq; il a la cons`equence suivante: Th´eor`eme. L’alg`ebre de cohomologie H ˚ pM, Qq est engendr´ee par les composantes de K¨unneth des classes de Chern de E. Let but de cette note est de montrer comment la m´ethode de la diagonale utlilis´ee dans [E-S] fournit une d´emonstration tr´es simple de ce th´eor`eme. Celcui-ci r´esulte de l’´enonc´e un peu plus g´en´eral que voice: * Avec

le support partiel du projet europe´een Science “Gremetry of Algebraic Varieties”, Contrat n˝ SCI-0398-C(A).

44

Sur la cohomologie de certains espaces de modules...

45

Proposition. Soient X une vari´et´e complex projective et lisse, et M 38 un espace de modules espace de modules de faisceaux stables sur x (par rapport a´ une polarisation fix´ee, cf. [M]). On fait les hypoth`eses suivantes: (i) La vari´et´e M est projective et lisse. (ii) Il existe un faiseceau de Poincar´e E sur M. (iii) Pour E, F dans M, on a Exti pE, Fq “ 0 pour i ě 2.

Alors l’alg`ebre de cohomologie H ˚ pM, Qq est engendr´ee par les composantes de K¨unneth des classes de Chern de E.

La d´emostration suit de pr`es celle du th. 1 de [E-S]. Rappelons-enl’ id`ee fondamentale: soit δ la classe de cohomologie de la diagonale dans H ˚ pM řˆ ˚M, Qq;˚ notons p et q les deux projections de ˆM sur M. Soit p µi ¨ q υi , la d´ecompostion de K¨unneth de δ; alors l’espace δ “ i

H ˚ pM, Qq est engendr´e par les vi . En effect, pour λ dans H ˚ pM, Qq, on a ÿ λ “ q˚ pδ ¨ p˚ λq “ degpλ ¨ µi qvi

d’o`u notre assertion. Il s’agint donce d’ exprimer la classe δ en fonction des classes de Chern du fibr´e universel. Notons p1 , p2 le deux projections de C ˆ M ˆ M sur C ˆ M, et π la projections sur M ˆ M; d´esignons par H le faisceau Hompp˚1 E, p˚2 Eq. Vul’,hypoth`ese (iii), l’hypercohomologie Rπ˚ H est repr`esent´ee dans la cat´egorie d´eriv´ee par un complexe de fibr´es K ‚ , nul en degr´e diff´erent de O rt 1. Autrement dit, il existe un morphisme de fibr´es u:K 0 ÝÑ K 1 tel qu, on ait, pour tout point x “ pE, Fq de M, une suite exacte upxq

0 Ñ HompE, Fq Ñ K 0 pxq ÝÝÑ k1 pxq Ñ Ext1 pE, Fq Ñ 0. Come l’espace HompE, Fq est non nul si et seulement si E et F est non nul si et seulement si E F sont isomorphes, on voit que la diagonale ∆ de M ˆ M co¨lncide ensemblistement avecle lieu de d´eg´en´erescence D de u (d´efine par l’annulation des mineurs de rang maximal de u). On 45

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Arnanud Beauville

peut prouver comme dans [E-S] l’ e´ galit´e sch´ematique, mais cela n’est pas n´ecessaire pour d´emontrer la proposition. Soit E un e´el´ement de M. On a rgpK 0 q ´ rgpK 1 q “ dim HompE, Fq ´ dim Ext1 pE, Fq 39

quel que soit le point pE, Fq de M ˆ M. Puisque Ext2 pE, Eq “ 0, la dimension m de M est e´gale a` dim Ext1 pE, Eq; ainsi la sous-vari´ee´ d´eterminantale D de MˆM a la codimension attendue rgpK 1 q´rgpk0 q` 1. Sa classe de cohomologie δ1 P H m pM ˆ M, Zq est alors donn´ee par la formule de Proteous δ1 “ cm pK 1 ´ K 0 q “ cm p´π!Hq, o`u π! d´esigne le foncteur image directe en K-th´eorie. Cette classe e´ tant multiple de la classe δ de la diagonale, on conclut avec le lemme suivant: ` de H ˚ pM, Qq engendr´ee par les Lemme. Soit A la sous-bQ-algbre composantes de K¨unneth des classes de Chern de E, et soient p et q les deux projecutionsřde M ˆ M sur calM. Les classes de Chern de π!H sont de la forme P˚ µi ¨ q˚ vi , avec µi , vi P A.

Notons r la projections de C ˆ M ˆ M sur C. Tout polynˆome en les classes de Chern de p˚1 calE et de P˚2 E est une somme de produits de la forme r˚ γ ¨ π˚ p˚ µ ¨ π˚ q˚ v, o`u µ et v appartiennent a` A. Le lemme r´emme r´sulte alors de la formule de Riemann-Roch chpπ!Hq “ π˚ pr˚ ToddpCq chpHqq. Remarque . La condition (iii) de la proposition est e´ videnmment tr`es con-traignante. Donnons deux exemples: a) X est une surface rationnelle ou r´egl´ee, et la polarisation H v´erifie H ¨K x ă 0. L’argument de [M, cor. 6.7.3] montre que la condition (iii) est satisfaite. Si de plus les coefficients ai du polynˆo me m de Hilbert ˆ des ˙ e´ l´ements de M, e´ crit sous la forme XpEq b H q “ 2 ř m`i ai , sont premiers entereux, les conditions (i) a` (iii) sont i i“0 satisfaites [M, §6]. 46

Sur la cohomologie de certains espaces de modules...

47

Dans le cas d’ une surface rationnelle, on obtient mieux. Pour toute vari´et´e T , d´esignons par CH ˚ pT q l’anneau de Chow de T ; grˆa ce a` l’isomorphisme CH ˚ pXtimesMq – CH ˚ pXq b CH ˚ pMq, on peut remplacer dans la d´emonstration de la proposition l’anneau de cohomologie par l’anneau de Chow. On en d´eduit quela cohomologie rationnelle de M est alg´ebrique, c’est-`a-dire que l’application “classe de cycles” de cycles” CH ˚ pMq b Q ÝÑ H ˚ pM, Qq est u isomphisme d’ anneaux. Dans le cas X “ P2 , ellingsrud et Strømme obtiennent le mˆome r´esultat sur Z, plus le fait que ces groupes sont sans torsion, grˆace a` l’outil suppl´ementaire de la suite spectrale de Beilinson. b) X est une vari´ete´e de Fano De dimension 3. Soit S une surface lisse appartenant au syst`eme lin´eaire | ´ K x | (de sorte que S est une surface K3). Lorsqu’elle est satisfaite, la condition (iii) a des 40 cons´equences remarquables [T]: elle entraˆıne que ”application de restriction E ÞÝÑ E|S d´efinit un isomorphisme de M une sous-vari´et´e largrangienne d’un espace de modules MS de fibr´es sur S (muni de sa structure symplectique canonique). Il me semble int´eressant de mettre en e´ vidence des espace de modules de fibr´es sur une vari´et´e de Fano (et d´ej`a sur P3 ) poss´edant la propri´et´e (iii).

R´ef´erences [A-B] M. Atiyah et R. Bott, Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. London A 308 (1982) 523–615. [E-S] G. Ellingsrud et S.A. Strømme, Towards the Chow ring of the Hilbert scheme of P2 , J. reine angew. Math. 441 (1993) 33–44. [M] M. Maruyama, Moduli of stable sheaves, II. J. Math. Kyoto Univ. 18 (1978) 557–614. [T] A. N. Tyurin, The moduli space of vector bundles on threefolds, surfaces and surves I, preprint (1990). 47

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Arnanud Beauville

Arnaud Beaville Universit´e Paris-Sud Math´ematiques- Bˆat. 425 91 405 Orsay Cedex, France

48

Some quantum analogues of solvable Lie groups C. De Concini, V.G. Kac and C. Procesi

Introduction 41

In the papers [DK1] [DK2], [DKP1] [DKP2]the quantized enveloping algebras introduced by Drinfeld and Jimbo have been studied in the case q “ ε, a primitive l-th root of 1 with l odd (cf. calx for basic definitions). Let us only recall for the moment that such algebras are canonically constructed starting from a Cartan matrix of finite type and in praticular we can talk of the associated classical objects (the root system, the simply connected algebraic group G. etc.) For such a algebra tha generic (resp. any) irreducible representation has dimesion equal to (resp. bounded by) lN where N is the number of positive roots and the set of irreducible representations has a canonical map to the big cell of the corresponding group G. In this paper we analyze the structure of some subalgebras of quanrized enveloping algebras corresponding to unipotent and solvable subgroups of G. These algebras have the non-commutative structure of iterated algebras of twisted polynomials with a derivation, an object which has often appeared in the general theory of non-commutative rings (see e.g. [KP], [GL] and references there). In pariticular, we find maximal demensions of their irreducible representations. Our results confirm the validity of the general philosophy that the representation theory is intimately connected to the Poisson geometry.

1 Twisted polynomial rings 1.1 In this section we will collect some well knownn definitions and properties of twisted derivations. Let A be an algebra and let σ be an automorphism of A. A twisted 49

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C. De Concini, V.G. Kac and C. Procesi

derivation of A realtive ot σ is a linear map D : A Ñ A such that: Dpabq “ Dpaqb ` σpaqDpbq. 42

Example . An element a P A induces an inner twisted derivation adσ a relative to σ defined by the formula: pabσ aqb “ ab ´ σpbqa. The following well-known fact is very useful in calculations with twisted derivations. (Hre and further we use “box” notation: „ rmsrm ´ 1s . . . rm ´ n ` 1s qn ´ q´n m rns “ , rns! “ r1sr2s . . . rns, “ n rns! q ´ q´1 One also writes rnsd , etc. if q is replaced by qd .) Proposition. Let a P A and let σ be an automorphism of A such that σpaq “ q2 a, where q is a scalar. Then „ m ÿ j jpm´1q m m p´1q q am´ j σ j pxqa j . padσ aq pxq “ j j“0

Proof. Let La and Ra denote the operators of left and right multiplications by a in A. Then adσ a “ La ´ Ra σ. Since La andRa commute, due to the assumption σpaq “ q2 a we have La pRa σq “ q´2 pRa σqLa . Now the proposition is immediate from the following well-known binomial formula applied to the algebra End A. Lemma. suppose that x andy y are elements of an algebra such that yx “ q2 xy for some scalar q. Then m „ ÿ m jpm´ jq j m´ j m px ` yq “ q xy . j j“0

50

51

Some quantum analogues of solvable Lie groups 43

Proof. is by induction on m using „ „ „ m`1 j m m m`1 q, “ q ` j j j´1 which follows from qb ras ` a´a rbs “ ra ` bs. ℓ1

Let ℓ be a positive integer and let q be a primitive ℓ-th root of 1. Let “ ℓ if ℓ is odd and = 12 ℓ if ℓ is even. Then, by definition, we have „ 1 ℓ “ 0 for all j such that 0 ă j ă ℓ1 . j

This together with Proposition 1.1 implies Corollary. Under the hypothesis of Proposition 1.1 we have: 1

1

1

1

padσ aqℓ pxq “ aℓ x ´ σℓ pxqaℓ if q is a primitive ℓ ´ th root of 1. Remark. Let D be a twisted derivation associated to an automorphism σ such that σD “ q2 Dσ. Then by induction on m one obtains the following well-known q-analogue of the Leibniz formula: m „ ÿ m jpm´ jq m´ j j m D pxyq “ q D pσ xqD j pyq. j j“0

1

It follows that if q is a primitive ℓ-th root of 1, then Dℓ is a twisted 1 derivation associated to σℓ 1.2 Given an automorphism σ of A and a twisted derivation D of A relative to σ we define the twisted polynomial algebra Aσ,D rxs in the indeterminate x to be the F-module A bF Frxs thought as formal polynomials with multiplication defined by the rule: xa “ σpaqx ` Dpaq. 51

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C. De Concini, V.G. Kac and C. Procesi

When D “ 0 we will also denote theis ring by Aσ rxs. Notice that the definition has been chosen in such a way that in the new ring the given twisted derivation becomes the inner derivation adσ x. Let us notice that if a, b P A and a is invertible we can perform the change of variables y :“ ax ` b and we see that Aσ,D rxs “ Aσ1 ,D1 rys. It is better to make the formulas explicit separately when b “ 0 and when a “ 1. In the fist case yc “ axc “ apσpcqx ` Dpcqq “ apσpcqqa´1 y ` aDpcq and we see that the new automorphism σ1 is the composition pAdaqσ, so that D1 :“ aD is a twisted derivation relative to σ1 . Here and further Ada stands for the inner automorphism: pAdaqx “ axa´1 . In the case a “ 1 we have yc “ px ` bqc “ σpcqx ` Dpbq ` bc “ σpcqy ` Dpbq ` bc ´ σpcqb, so that D1 “ D ` adσ b. Summarizing we have Proposition. Changing σ, D to pAdaqσ, aD (resp. to σ, D ` Db ) does not change the twisted polynomial ring up to isomorphism. We may express the previous fact with a definition: For a ring A two pairs pσ, Dq and pσ1 , D1 q are equivalent if they are obtained one from the other by the above moves. If D “ 0 we can also consider the twisted Laurent polynomial algebra Aσ rx, x´1 s. It is clear that if A has no zero divisors, then the algebras Aσ,D rxs and Aσ rx, x´1 s also have no zero divisors. The importance for us of twisted polynomial algebras will be clear in the section on quantum groups. 1.3 We want to study special cases of the previous construction. Let us first consider a finite dimensional semisimple algebra. A over À and algebraically closed field F, let i Fei be the fixed points of the center of A under σ where the ei are central idempotents. We have Dpei q “ Dpe2i q “ 2Dpei qei hence Dpei q “ 0 and, À if x “ xei , then Dpxq “ Dpxqei . It follows that, decomposing A i Aei , each compo52

Some quantum analogues of solvable Lie groups

53

nent Aei is stable under σ and D and thus we have à Aσ,D rxs “ pAei qσ,D rxs. i

This allows us to restrict our analysis to the case in which 1 is the only fixed central idempotent. The second special case is described by the following: Lemma. Consider the algebra A “ F‘k with σ the cyclic permutation of the summands, and let D be a twisted derivation of this algebra relative to σ. Then D is an inner twisted derivation. Proof. Compute D on the idempotents: Dpei q “ Dpe2i q “ Dpei qpei ei`1 q. Hence we must have Dpei q “ ai ei ´ bi ei`1 and from 0 Dpei ei`1 q “ Dpei qei`1 Dpei`1 q we deduce bi “ ai`1 . Let now a pa1 , a2 , . . . , ak q; an easy computation shows that D “ adσ a.

` 45 “ “

Proposition. Let σ be the cyclic permutation of teh summands of the algebra F‘k . Then “ ´1 ‰ (a) F‘k σ“ x, x ‰ is an Azumaya algebra of degree k over its center F xk , x´k . “ ´1 ‰ “ ‰ bFrxk ,x´k s F x, x´1 is the algebra of k ˆ k matri(b) R :“ F‘k σ “x, x ‰ ces over F x, x´1 .

Proof. It is enough to prove (b). Let u :“ x b x´1 , ei :“ ei b 1; we have uk “ xk b x´k “ 1 and uei “ ei`1 u. From these formulas it easily follows that the elements ei u j pi, j “ 1, . . . , kq span a subalgebra A and that there exists an isomorphism AĄ ÝÑpFq mapping F‘k to the diagonal matrices “ ´1and ‰ u to the matrix of the cyclic permutation. Then R “ A bF F x, x . 1.4 Assuem now that A is semi-simple and that σ induces a cyclic permutation of the central idempotents.

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C. De Concini, V.G. Kac and C. Procesi

Lemma. (a) A “ Md pFq‘k (b) Let D be a twisted derivation of A realtive to σ. Then the pair pσ, Dq is equivalent to the pair pσ1 , 0q where σ1 pa1 , a2 , . . . , ak q “ pak , a1 , a2 , . . . , ak´1 q

(1.4.1)

Proof. Since σ permutes transitively the simple blocks they must all have the same degree d so that A “ Md pFq‘k . Furthermore we can arrange the identifications of the simple blocks with matrices so that: σ1 pa1 , a2 , . . . , ak q “ pτpak q, a1 , a2 , . . . , ak´1 q,

46

where τ is an automorphism of Md pFq. Any such automorphism in inner, hence after composing σ with an inner automorphism, we any assume in the previous formula that τ “ 1, Then we think of A as Md pFq b F‘k , the new automorphism being of the form 1 b σ1 where σ1 : F‘k Ñ F‘k is given by (1.4.1). We also have that Md pFq “ Aσ and F‘k is the centralizer of Aσ . Nest observe that D restricted to Aσ is a derivation of Md pFq with values in ‘ki“1 Md pFq, i.e., Dpaq “ pD1 paq, D2 paq, . . . , Dk paqq where each Di is a derivation of Md pFq. Since for Md pFq. all derivations are inner we can find an element u P A such that Dpaq “ ru, as for all a P Md F. So pD ´ adσ uqpaq “ ru, as ´ pua ´ σpaquq “ 0 for a P Aσ . Thus, changing D by adding ´adσ u we may assume that D “ 0 on Md pFq. Now consider b P F‘k and ac P Md pFq; we have Dpbqa “ Dpbaq “ Dpabq “ aDpbq. Since F‘k is the centralizer of Md pFq we have Dpbq P F‘k and D induces a twisted derivation of F‘k . By Lemma 1.3 this last derivation is inner and the claim is proved. Summarizing we have Proposition. Let A be a finite- dimensional semisimple algebra over an algebraically closed field F. Let σ be an automorphism of A which induces a cyclic permutation of order k of the central idempotents of A. Let D be a twisted derivation of A relative to σ. Then: Aσ,D rxs – Md pFq b F‘k σ rxs, 54

Some quantum analogues of solvable Lie groups

55

´1 Aσ,D rx, x´1 s – Md pFq b F‘k σ rx, x s.

This last algebra is Azumaya of degree dk. 1.5 We can now globalize the previous constructions. Let A be a prime algebra (i.e. aAb “ 0, a, b P A, implies that a “ 0 or b “ 0) over a field F and let Z be the center of A. Then Z is a domain and A is torsion free module over Z. Assume that A is a finite module over Z. Then A embeds in a finite-dimensional central simple algebra QpAq “ A bZ QpZq, where QpZq is the ring of fractions of Z. If QpZq denotes the algebraic closure of QpZq is the ring of fractions of Z. If QpZq denotes the algebraic closure of QpZq in the ring of fractions of Z. If QpZq denotes the algebraic closure of QpZq we have that A bz QpZq is the full ring Md pQpZqq of d ˆ d matrices over QpZq. Then d is called the degree of A. Let σ be an automorphism of the algebra A and let D be a twisted derivation of A relative to σ. Assume that (a) There is subalgebra Z0 of Z, such that Z in finite over Z0 . (b) D vanishes on Z0 and σ restricted to Z0 is the identity. These assumptions imply that σ restricted to Z is an automorphism of finite order. Let d be the degree of A and let k be the order of σ on the center Z. Assume that the field F has characteristic 0. The main result 47 of this section is: Theorem. Under the above assumptions the twisted polynomial algebra Aσ,D rxs is an order in a central simple algebra of degree kd. Proof. Let Z σ be the fixed points in Z of σ. By the definition, it is cleat that D restricted to Z σ is derivation. Since it vanishes on a subalgebra over which it is finite hence algebraic and since we are in characteristic zero it follows that D vanishes on Z σ . Let us embed Z σ in an algebraically closed field L and let us consider the algebra A bZ σ L “ L‘k and A ˆZ L “ Md pLq. Thus we get that A bZ σ L “ ‘ki“1 Md pLq. The 55

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pair σ, D extends to A bZ σ L and using the same notations we have that pA bZ σ Lqσ,D rxs “ pA sigma,D rxsq bZ σ L. We are now in the situation of a semisimple algebra which we have already studied and the claim follows. Corollary. Under the above assumptions, Aσ,D rxs and Aσ rxs have the same degree. Remark. The previous analysis yields in fact a stronger result. Consider the open set of Spec Z where A is an Azumaya algebra; it is clearly σstable. In it we consider the open part where σ has order exactly k. Every orbit of k elements of the group generated by σ gives a point Fppq in Spec Z σ and A bZ Z bZ σ Fppq “ ‘ki“1 Md pFppqq. Thus we can apply the previous theory which allows us to describe the fiber over Fppq of the spectrum of Aσ,D rxs. 1.6 Let A be a prime algebra over a field F of characteristic 0, let x1 , . . . , xn be a set of generators of A and let Z0 be a central subalgebra of A. For each i “ 1, . . . , K, denote by Ai the subalgebra of A generated by x1 , . . . , xi and let Z0i “ Z0 X Ai . We assume that the following three conditions hold for each i “ 1, . . . , k: (a) xi x j “ bi j x j xi ` Pi j if i ą j. where bi j P F, Pi j P Ai´1 . (b) Ai is a finite module over Z0i . (c) Formulas σi px j q “ bi j x j for j ă i define a automorphism of Ai´1 which is the identity on Z0i´1 .

48

Note that letting Di px j q “ Pi j for J ă i, we obtain Ai “ Ai´1 σi ,Di rxi s, so that A is an iteratated twisted polynomial algebra, Note also that each triple pAi´1 , σi , Di q satisfies assumptions 1.5 (a) and (b). i We may consider the twisted polynomial algebras A with zero derivations, so that the relations are xi x j “ bi j x j xi for j ă i. We call this the associated quasipolynomial algebra (as in [DK1]). We can prove now the main theorem of this section. 56

Some quantum analogues of solvable Lie groups

57

Theorem. Under the above assumptions, the degree of A is equal to the degree of the associated quasipolynomial algebra A. Proof. We use the following remark. If there is an index h such that the element Pi j “ 0 for all i ą h and all j, then monomials in the variables different from xh form as subalgebra B and the algebra A is a twisted polynomial ring Bσ,D rXh s. The associated ring Bσ rXh s is obtained by setting ph j “ 0 for all j. Having made this remark we see that can inductively modify the relations 1.6(a) so that at the h-th step we have an algebra Anh with the same type of relations but Pi j “ 0 for all i ą n´h and all j. Since Anh and Anh´1 are of type Bσ,D rxs and Bσ rxs respectively we see, by Corollary 1.5, that they have all the same degree.

2 Quantum groups 2.1 Let pai j q be an indecomposable n ˆ n Cartan matrix and let d1 , . . . , dn be relatively prime positive integers such tha di ai j “ d j a ji . Recall the associated notions of the weight, coroot and root lattices p, Q_ and Q, of the root and coroot systems R and R_ , of the Weyl group W, the W-invariant bilinear form p.|.q, etc.: Let P be a lattice over Z with basis ω1 , . . . , ωn and let Q_ “ HomZ _ _ _ pP, Zq be the dual řnlattice with the dual basis α1 , . . . , αn , i.e. xωi , αn y “ δi j . Let P` “ i“1 Z` ωi . Let ρ“ řn

n ÿ

ωi ,

i“1

αj “

n ÿ

i“1

ai j ωi p j “ 1, . . . , nq,

ř and let Q “ j“1 Zα j Ă P, and Q` “ nj“1 Z` α j . Define automorphisms si of p by si pω j q “ ω j ´δi j α j pi, j “ 1, . . . , nq. Then si pα j q “ α j ´ ai j αi . Let W be the subgroup of GLppq generated by s1 , . . . , sn . Let ( _ Π “ tα1 , . . . , αn u, Π_ “ α_ 1 , . . . , αn , R “ WΠ,

R` “ R X Q ` , 57

R_ “ WΠ_ .

58 49

C. De Concini, V.G. Kac and C. Procesi

The map αi ÞÝÑ α_ i extends uniquely to a bijective W-equivariant map _ α ÞÝÑ αi between R and R_ . The reflection sα defined by sα pλq “ λ ´ xλ, α_ yα lies in W for each α P R, so that sαi “ si . Define a bilinear pairing P ˆ Q Ñ Z by pωi |α j q “ δi j d j . Then pαi |α j q “ di ai j , giving a symmetric Z-valued W-invariant bilinear form on Q such that pα|αq P 2Z. We may indentify Q_ with a sublattice of the Q-span of P (containing Q) using this form. Then: ´1 _ α_ i “ di αi , α “ 2α{pα|αq.

One defines the simply connected quantum“ group U associated to ‰ the matrix pai j q as analgebra over the ring A :“ q, q´1 , pqdi ´ q´di q´1 . with generators Ei , Fi pi “ 1, . . . , nq, Kα pα P Pq subject to the following relations (this is simple variation of the construction of Drinfeld and Jimbo): Kα Kβ “ kα`β , k0 “ 1,

σα pEi q “ qpα|αi q Ei , σα pFi q “ q´pα|αi q Fi, kα ´ K´αi rEi , F j s “ δi j di , q i ´ q´di padσ´αi Ei q1´ai j E j “ 0, padσαi Fi q1´ai j F j “ 0 pi ‰ jq, where σα “ Ad Kα . Recall that U has a Hopf algebra structure with comultiplications ∆, antipode S and counit η defined by: ∆Ei “ Ei b 1 ` Kαi b Ei , ∆Fi “ Fi b K´αi ` 1 b Fi , ∆kα “ Kα b Kα ,

S Ei “ ´K´αi Ei , S Fi “ ´Fi Ki , S kα “ K´α , ηEi “ 0, ηFi “ 0, ηKα “ 1.

Recall that the braid group BW (associated to W), whose canonical generators one denotes buy T i , acts as a group of automorphisms of the algebra U ([L]):

T i Kα “ K si pαq , T i Ei “ ´Fi Kαi 58

Some quantum analogues of solvable Lie groups Ti E j “

59

1 !padσαi p´Ei qq´ai j E j , r´ai j sdi

T i k “ kT i ,

where k is a conjugate-linear anti-automorphism of U, viewed as an algebra over IC, defined by: kEi “ Fi , kFi “ Ei , kKα “ Kα , kq “ q´1 . 2.2 Fix a reduced expression ω0 “ S i1 . . . siN of the longest element of W, and let β1 “ αi1 , β2 “ si1 pαi2 q, . . . , βN “ si1 . . . siN´1 pαiN q be the corresponding convex ordering of R` . Introduce the correspond- 50 ing root vectors pm “ 1, . . . , Nq ([L]): Eβm “ T i1 . . . , T im´1 Eim , Eβm “ T i1 . . . T im´1 Fim “ kEβ (they depend on the choice of the reduced expression). N we let For k “ pk1 . . . , kN q P Z` E k “ Eβk11 . . . EβN kN , F k “ kE k . N , α P P, from Lemma. (a) rLs The elements F k Kα E r , where k, r P Z` a basis of U over A.

(b) rLS s For i ă j one has: Eβi Eβ j ´ qpβi |β j q Eβ j Eβi “

ÿ

N kPZ`

ck E k ,

(2.2.1)

‰ “ where ck P IC q, q´1 and ck ‰ 0 only when k “ pk1 , . . . , kN q is such that k s “ 0 for s ď i and s ě j. An immediate corollary is the following: Let w “ si1 . . . sik which we complete to a reduced expression ω0 “ si1 . . . siN of the longest element of W. Consider the elements Eβ j , j “ 1, . . . , k. Then we have: 59

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Proposition. (a) The elements Eβ j , j “ 1, . . . , k, generate a subalgebra U w which is independent of the choice of the reduced expression of w. (b) If w1 “ ws with s a simple reflection and lpw1 q “ lpwq ` 1 “ 1 w rE k`1, then U w is a twisted polynomial algebra of type Uσ,D βk`1 s, where the formulas for σ and D are implicitly given in the formulas (2.2.1). 51

Proof. (a) Using the face that once can pass from one reduced expression of w to another by braid relations one reduces to the case of rank 2 where one repeats the analysis made by Luszting ([L]). (b) is clear by Lemma 2.2. The elements Kα clearly normalize the algebras U w and when we add them to these algebras we are performing an iterated construction of Laurent twisted polynomials. The resulting algebras will be called Bw . Since the algebras U w and Bw are iterated twisted polynomial rings with relations of the type 1.6(a) we can consider the associated quasipolyw w nomial alagebras, and we will denote them by U and B . Notice that the latter algebras depend on the reduced expression chosen for w. Of course the defining relations for these algebras are obtained from (2.2.1) by replacing the right-hand side by zero. We could of coures also perform the same construction with the negative roots but this is not strictly necessary since we can simply apply the anti-automorphism k to define the analogous negative objects.

3 Degrees of algebras UEw and BwE 3.1 We specialize now the previous sections to the case q “ E, a primitive ℓ-th root of 1. Assuming that ℓ1 ą max di . we may consider the i

specialized algebras:

UE “ U{pq ´ Eq, UEw “ U w {pq ´ Eq, BwE “ Bw {pq ´ Eq, etc. We have obvious subalgebra inclusions UEw Ă BwE Ă UE . 60

61

Some quantum analogues of solvable Lie groups

First, let us recall and give a simple proof of the following crucial fact [DK1]: Proposition. Elements Eαℓ pα P Rq and Kβℓ pβ P Pq lie in the centre zE of UE if ℓ1 ą max |ai j | (for any generalized Cartan matrix pai j qq. i, j

Proof. The only non-trivial thing to check is that rEiℓ , E j s “ 0 for i ‰ j. 1 From the “Serre relations” it is immediate that padσ´αi Ei qℓ E j “ 0. Due to Corollary 1.1, this can be rewritten as 1

1

1

Eiℓ E j “ E´ℓ pαi |α j q E j Eiℓ , proving the claim.

As has been alreadu remarked, the algebras UEw and BwE are iterated twisted polynomial algebras with relations of the type 1.6(a) Proposition 3.1 shows that they satisfy conditions 1.6(b) and (c). Hence Theorem 1.6 52 implies w

w

Corollary. Algebras UEw and U E (resp.BwE and BE ) have the same degree. w

w

3.2 We proceed to calculate the degrees of algebras U E and BE . Recal that these algebras are, up to inverting some variables, quasipolynomial algebras whose generations satisfy relations of type xi x j “ bi j x j xi , i, j “ 1, . . . , s, where the elements bi j have tha special form bi j “ Eci j , the ci j being entries of a skew-symmetric integral s ˆ s matrix H. As we have shown in [[DKP2], Proposition 2.2] considering H as the mas , the degree of the corresponding trix of a linear map Z s Ñ pZ{pℓqq ? twisted polynomial algebra is h, where h is the number of element of the image of this map. Fix w P W and its reduced expression w “ si1 . . . sik . We shall w w denote the matrix H for the algebras U E and BE by A and S respectively. First we describe explicitly these matrices. Let d ““ 2 unless pai j q is of type G2 in which case d “ 6, and ‰ 1 ´1 let Z “ Z d . Consider the roots β1 , . . . βk as in Section 2.2, and 61

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consider the free Z1 -module V with basis u1 , . . . , uk . Define on V a skewsymmetric bilinear form by xui |u j y “ pβi |β j q if i ă j. Then A is the matrix of this bilinear form in the basis tui u. Identifying V with its dual V ˚ using the give basis, we may think of A as a linear operator from V to itself. Furthermore, ˙ ˆ A ´t C S “ C 0 where C is the n ˆ k matrix ppωi |β j qq1ďiďn,1ď jďk . We may think of the matrix C as a linear map from the module V with the basis u1 , . . . , Uk to _ the module Q_ bZ Z1 with the basis α_ 1 , . . . , αn . Then we have: Cpui q “ βi , i “ 1, . . . , k.

53

(3.2.1)

To study the matrices A and S we need the following ř Lemma. Given ω “ ni“1 δi ωi with δi “ 0 or 1, set Iω “ tt P 1, . . . , k|sit pωq ‰ ωu.

Then wpωq “ ω ´

ÿ

β j.

jPiw

Proof. by induction on the length of w . Write w “ w1 sik . If k R Iω then wpωq “ w1 pωq and we are done by induction. Otherwise wpωq “ w1 pω ´ αik q “ w1 pωq ´ βk and again we are done by induction. Note that 1, 2, . . . , k “

n š

I ωi .

i“1

3.3 Consider the operators: M “ pA ´ t Cq and N “ pCOq so that S “ M ‘ N. 62

Some quantum analogues of solvable Lie groups

63

Lemma. (a) The operator M is surjective. `ř ˘ (b) The vectors vω :“ tPIw ut ´ ω ´ wpωq, as ω runs through the fundamental weights, form a basis of the kernel of M. ř (c) Npvω q “ ω ´ wpωq “ tPIω βt . Proof. (a) We have by a straightforward computation: ÿ S pui ` βi q “ ´pβi |βi qui ´ 2 pβi |β j qu j ´ βi , jąi

and Mpui ` βi q “ ´pβi |βi qui ´ 2

ÿ

pβi |β j qu j

jąi

Since pβi |βi q is invertible in Z1 the claim follows. (b) Since the vectors vω are part of a basis and, by (a), the kernle of M is a direct summand, it is enough to show that these vectors lier in the kernel. Now to check that Mpvωi q “ 0 is equivalent to seeing tha vωi lies in the kernel of the corresponding skew-symmetric form, i.e. xu j |vωi y “ 0 for all j “ 1, . . . , k: Using Lemma 3.2, we have ÿ xu j |vωi y “ ´2 pβ j |βt q ` 2pβ j |wpωi qq ` a j , (3.3.1) tą j

54 where a j “ 0 if j R Iωi and a j “ pβ j |β j q otherwise. We proceed by inductions on k “ lpwq. Let us write vωi pwq to stress the dependence on w. For k “ 0 there is nothing to prove. Let w “ w1 sik with lpw1 q “ lpwq ´ 1. We distinguish two cases according to whether i “ ik or not. If i “‰ ik , i.e. k R Iωi , we have that vωi “ vωi pw1 q hence the claim follows by induction if j ă k. For j “ k we obtain from (3.3.1):

xuk |vωi y “ ´2pβk |wpωi qq “ ´2pw1 pαik q|w1 pωi qq “ ´2pαik |ωi q “ 0. Assume now that ik “ i so that w “ w1 si . Then vωi pwq “ vωi pw1 q ` uk ´ βk . For j ă k by induction we get: xu j |vωi y “ xu j |uk y ´ xu j |βk y “ ´pβ j |βk q ` pβ j |βk q “ 0. 63

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C. De Concini, V.G. Kac and C. Procesi

Finally if j “ k we have: 2pβk |wpωi qq ` pβk |βk q “ 2pw1 αi |w1 pωi ´ αi qq ` pαi |αi q

“ 2pαi |ωi q ´ pαi |αi q “ 0. ř Finally using (3.2.1), we have: Npvωi q “ tPIω βt , hence (c) follows i form Lemma 3.2. 3.4 In order to compute the kernel of S we need to compute the kernel of N on the submodule spanned by the vectors vωi . Let us identify this module with the weight lattice p by identifying vωi with ωi . By Lemma 3.3(c), we see that N in identified with map 1 ´ w : P Ñ Q. At this point we need the following fact: řn Lemma. Consider the highest root θ “ i“1 ai αi of the root system R. ‰ “ ´1 ´1 1 1 2 Let Z “ Z a1 , . . . , an , and let M “ M bZ Z1 , M 2 “ M bZ Z2 for M “ p or Q. Then for any w P W, the Z2 -submodule p1 ´ wqP2 of Q2 is a direct summand.

55

Proof. Recall that one can represent w in the form w “ S γ1 . . . sγm where γ1 . . . γm is a linearly independent set of roots (see e.g. [C]). Since ř one of the ri is 1 or 2, it follows that in the decomposition γ_ “ i ri α_ i 1 1 p1 ´ sγ qP “ Zγ . Since 1 ´ w “ p1 ´ sγ1 . . . sγm´1 qsγm ` p1 ´ sγm q, we deduce by induction that p1 ´ wqP1 “

m ÿ

Z 1 γm

(3.4.1)

i“1

Recall now that any sublattice of Q spanned over Z by some roots is a Z-span of a set of roots obtained from Π by iterating the following procedure: add a highest root to the set of simple roots, then remove several other roots form this set. The index of the lattice M thus obtained in M bZ Q X Q is equal to the product of coefficients of removed roots in the added highest root. Hence it follows from (3.4.1) that pp1 ´ wqP2 q bZ Q X Q2 “ p1 ´ wqP”, proving the claim.

64

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Some quantum analogues of solvable Lie groups

We call ℓ ą 1 a good integer if it is relatively prime to d and to all the ai . Theorem. If ℓ is a good integer, then w

1

deg BwE “ deg BE “ ell 2 pℓpwq`rankp1´wqq . w

Proof. From the above descussion we see that deg BE “ ℓ s , where s “ pℓpwq`nq´pn´rankp1´wqq, which together with Corollary 3.1 proves the claim. 3.5 We pass now to UEw . For this we need to compute the image of the matrix A. Computing first its kernel, ř we have that K erř A is identified with the set of linear combinations c v for which i i ωi i ci pωi ` ř wpωi qq “ 0 i.e. i ci ωi P kerp1 ` wq. This requires a case by case analysis. A simple case is when w0 “ ´1, so that 1 ` w “ w0 p´1 ` w0 wq and one reduces to the previous case. Thus we get Proposition. If w0 “ ´1 (i.e. for types different from An , D2n`1 and E6 ) and if ℓ is a good integer, we have: w

1

deg UEw “ deg U E “ ℓ 2 pℓpwq`rankp1`wq´nq . Let us note the special case w “ w0 . Remark that defining t ω :“ ´w0 pωq we have an involution ω Ñ t ω on the set of fundamental weights. let us denote by s the number of orbits of this involution. Theorem. If E is a primitive ℓ-th root of 1, where ℓ is an integer greater 56 than 1 and relatively prime to d, then algebra UEw0 and BwE 0 have degrees ℓ f racN´s2 and ℓ

N`s 2

respectively.

Proof. In this case lpw0 q “ N and the maps ω Ñ ω ` w0 pωq and ω Ñ ω ´ w0 pωq are ω Ñ ω ´ t ω and ω Ñ ω ` t ω and so their ranks are clearly n ´ s and s respectively. 65

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C. De Concini, V.G. Kac and C. Procesi

4 Poisson structure 4.1 Before we revert to the discussion of our algebras we want to make a general remark. Assume that we have a manifold M and a vector bundle V of algebras with 1 (i.e. 1 and the multiplication map are smooth sections). We identify the functions on M with the sections of V which are multiples of 1. Let D be a derivation of V, i.e. a derivation of the algebra of sections which maps the algebra of functions on M into itself and let X be the corresponding vector field on M. Proposition. For each point p P M there exists a neighborhood U p and a map ϕt defined for |t| sufficiently small on V|U p which is a morphism of vector bundles covering the germ of the 1-parameter group generated by X and is also an isomorphism of algebras. Proof. The hypotheses on D imply that it is a vector field on V linear on the fibers, hence we have the existence of a local lift of the 1-parameter group as a morphism of vector bundles. The condition of being a derivation implies that the lift preserves the multiplications section i.e. it is a morphism of algebras. We will have to consider a variation of this: suppose M is a Poisson manifold and assume furthermore that the Poisson structure lifts to V i.e. for each (local) functions f and section s we have a Poisson bracket which is a derivation. This means that we have a lift of the Hamiltonian vector fields as in the previous proposition. We deduce: Corollary. Under the previous hypotheses, the fibers of V over points of a given symplectic leaf of M are all isomorphic as algebras. Proof. The proposition implies that in a neighborhood of a point in a leat the algebras are isomorphic but since the notion of isomorphism is transitive this implies the claim. 57

4.2 Let us recall some basic facts on Poisson groups (we refer to [D], [STS], [LW] for basic definitions and properties). Since a Poisson structure on a manifold M is a special type of a section of Λ2 T pMq it can be 66

Some quantum analogues of solvable Lie groups

67

viewed as a linear map from T ˚ pMq to T pMq. The image of this map is thus a distribution on the manifold M, It can be integrated so that we have a decomposition of M into symplectic leaves which are connected locally closed submanifolds whose tangent spaces are the spaced of the distribution. In fact in our case the leaves will turn out to be Zarisko open sets of algebraic of algebraic subvarieties. For a group H the tangent space at each point can be identified to the Lie algebra ~ by left translation and thus a Poisson structure on H can be given as a family of maps γh : ~˚ Ñ ~ as h P H. Let G be an algebraic group and H, K Ă G algebraic groups if their corresponding Lie non-algebras pg, ~, kq form a Manin triple , i.e. (cf. [D], [LW]) if g has a non-degenerate,symmetric invariant bilinear form with respect to which the Lie subalgebras ~ an k are isotropic and g “ ~ ‘ k (as vector spaces). Then it follows that we have a canonical isomorphism ~˚ “ k. Having identified ~˚ with k, the Poisson structure on H is thus described by giving for every h P H a linear map γh : k Ñ ~. Let x P k, consider x as an element of g, set π : g Ñ ~ to be the projection with kernel k. Set: γh pxq “ pAdhqπ pAdhq´1 pxq. Then one can verify (aw in [LW]) that the corresponding tensor satisfies the required properties of a Poisson structure. (In fact any Poisson structure on H can be obtained in this way.) Notice now that the (restriction of the) canonical map: δ : H Ñ G{K is an e´ tale covering of some open set is G{K. Thus for every point h P H we can identify the tangent space to H in h with the tangent space of G{K at deltaphq. By using right translation we can then identify the tangent space to G{K at δphq with g{pAdhqK, the tangent space at h P H with ~ by right translation and the isomorphism between them with the projection ~ Ñ g{pAdhqk. Using all these identifications once verifies that the map γh previously considered is the map induced by differentiating the left K-action on G{K. From this it follows. 67

68

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C. De Concini, V.G. Kac and C. Procesi

Proposition. The symplectic leaves for the symplectic structure on H coincide with the connected components of the preimages under δ of K-orbits under the left multiplications of G{K. Consider now a quotient Poisson groups S and H, that is S is a quotient group of H and the ring ICrS s Ă ICrHs is a Poisson subalgebra. Let U be the kernel of the quotient homomorphism ϕ : H Ñ S , let s “ Lie S , u “ Lie U and dϕ : ~ Ñ s the Lie algebra quotient map. Then u is an ideal in ~ and we identify s˚ with a subspace of ~˚ “ k by taking uK Ă g under the invariant form and intersecting it with k. Then for p P S the linear mar: γ p : s˚ Ñ s giving rise to the Poisson structure is given by: γ p “ pdϕq ¨ pγ p |s˚ q where tildep P H is any representative of p(γ p is independent of the choice of p). ˜ The construction of the Manin triple corresponding to the Poisson manifold S is obtained from the following simple fact: Lemma. Let pg, ~, kq be a Manin triple of Lie algebras, and let u Ă ~ be an ideal such that uK pingq intersected with k is a subalgebra of the Lie algebra k. Then (a) uK is a subalgebra of g and u is an ideal of u K. (b) puK {u, ~{u, k X uK q is Manin triple, where the bilinear form on uK is induced by that of g. Proof. Straightforward.

4.3 In the remaining sections we will apply the above remarks to the Poisson groups associated to the Hopf algebra UE and its Hopf subalgebra BE :“ BwE 0 , and will derive some results on representations of the algebra BE . From now on E is a primitive ℓ-th root of 1 where ℓ ą 1 is relatively prime to d. Let Z0 (resp. Z0` ) be the subalgebra of UE (resp. BE ) generated by the elements Eαℓ with α P R (resp. α P R` ) and Kβ with β P P. (We 68

Some quantum analogues of solvable Lie groups

69

assume fixed a reduced expression of w0 ; Z0 and Z0` are independent of this choice [DK1].) Recall that they are central subalgebras (Proposition 3.1). It was shown in [DK1] that Z0 and Z0` are Hopf subalgebras, hence Spec Z0 and Spec Z0` have a canonical structure of an affine algebraic group. Furthermore. since UE is a specialization of the algebra U at q “ E, the center ZE of UE possesses a canonical Poisson bracket given by the formula: “ ‰ aˆ , bˆ ta, bu “ 2 mod pq ´ Eq, a, b P ZE , 2ℓ pq ´ Eq where aˆ denotes the preimage of a under the canoncila homomorphism 59 U Ñ UE . The algebras Z0 and Spec Z0` have a canonical structure of Poisson algebraic groups, Spec Z0` begin a quotient Poisson group of Spec Z0 . In [DKP1] an explicit isomorphism was constructed between the Poisson grout Spec Z0 and a Poisson group H which is described below. We shall identify these Poisson groups. Let G be the connected simply connected algebraic group associated to the Cartan Matrix pai j q and let g be its (complex) Lie algebra. We fix the triangular decomposition g “ u´ ` t ` U` , let b˘ “ t ` u˘ , and denote by p.|.q the invariant bilinear form on g which on the set of roots R Ă t˚ coincides with that defined in Section 2.1. Let U˘ , B˘ and T be the algebraic subgroups of G corresponding to Lie algebras u˘ , B˘ and t. Then as an algebraic group, H is the following subgroup of G ˆ G: ( H “ ptu` , t´1 u´ q|t P T, u˘ P U˘ . The Poisson structure on H is given by the Manin triple pg ‘ g, ~, kq, where ~ “ tpt ` u` , ´t ` u´ q|t P t, u˘ P u˘ u, k “ tpg, gq|g P gu ,

and the invariant bilinear form of g ‘ g is ppx1 , x2 q|py1 , y2 qq “ ´px1 |y1 q ` px2 |y2 q. 69

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We identify the group B` “ H{tp1, u´ |u´ P U´ qu. The Manin triple generating Poisson structure on B` is obtained from pa‘g, ~, kq by taking the ideal u “ tp0, u´ , u´ P u´ qu and applying the construction given by Lemma 4.2. We clearly obatina the triple pq‘t, b` , b´ q, where we used identifications b˘ “ tpu˘ ´ t, ˘tq|u˘ P u˘ , t P tu. According to the general recipe of Proposition 4.2, the symplectic leaves of the Poisson group B` are obtained as follows. We identify the groups B˘ with the following subgroups of G ˆ T : ( B˘ “ pt´1 u˘ , t˘1 q|t P T, u˘ P U˘ . The inclusion B` Ă G ˆ T induces an e´ tale morphism δ : B` Ñ pG ˆ T q{B´ . 60

Then the symoplectic leaves of B` are the connected components of the preimages under the map under the map δ of B´ -orbits on G ˆ T {B´ under the left multiplication . In order to analyze the B´ -orbits on g ˆ T {B´ , let µ˘ : B˘ Ñ T denote the canonical homomorphisms with kernels U˘ and consider the equivariant isomorphism of B´ -varieties γ : G{U´ Ñ pG ˆ T q{B´ given by γpgU´ q “ pg, 1qB´ , where B´ acts on G{U´ by bpgU´ q “ bgµ´ pbqU´ .

(4.3.1)

Then the map δ gets identified with the map δ : B` Ñ G{U´ given by δpbq “ bµ`pbqU´ . We want to study the orbits of the action (4.3.1) of B´ on G{U´ . Consider the action of B´ on G{B´ by left multiplication. Then the canonical map pi : G{U´ Ñ G{B´ is B´ -equivariant, hence π maps every B´ -orbit O in G{U´ to a B´ -orbit in G{B´ , i.e. a Schubert cell Cw “ b´ wB´ {B´ for some e P W. We shall say that the orbit O is associated to w. 70

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Remark. We have a sequence of maps: δ

γ´1

π

B` Ý Ñ pG ˆ T q{B´ ÝÝÑ G{U´ Ý Ñ G{B` . Let ψ “ π ˝ γ´1 ˝ δ and Xw “ B` X B´ wB´ . Then: π´1 pCw q “ b´ wB´ {U´ and ψ´1 pCw q “ Xw .

We can prove now the following Proposition. Let O be a B´ -orbit in G{U´ under the action (4.3.1) associated to w P W. Then the morphism: π|O : O Ñ Cw is a principal torus bundle with structure groups:

In particular:

( T w :“ w´1 ptqt´1 , where t P T .

dim O “ dim Cw ` dim T w “ lpwq ` rankpI ´ wq. Proof. For g P G we shall write rgs for the coset g U´ . The morphism π 61 is clearly a principle T -bundle with T acting on the right by rgst :“ rgts. The action (4.3.1) fo B´ -orbits. Each of B´ commutes with the right T -action so that T permutes the B´ -orbits. Each B´ -orbit is a principal bundle whose structure group is the subtorus of T which stabilizes the orbit. This subtorus is independent of the orbit since T is commutative. In order to compute it we proceed as follows. Let rg1 s, rg2 s be two elements in O mapping to w P Cw . We may assume the g1 “ nh, g2 “ nk with h, k P T uniquely determined, where n P NG pT q is representative of w. Suppose that brnhs “ brnks, b P B´ . We can first reduce to the case b “ t P T ; indeed, writing b “ ut we see that u must fix w P Cw hence un “ nu1 with u1 P U´ and hence u acts trivially on trnhs. Next we have that, by definition of the T -action (4.3.1), “ ‰ “ ‰ rnks “ tnht´1 “ npn´1 tnht´1 q ` ˘ ` ˘ hence k “ n´1 tnht´1 or k “ h h´1 n´1 tnht´1 “ h n´1 tnt´1 as required. 71

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Lemma. Let O Ă B` be a symplectic leaf associated to w. Then OT “ Xw . Proof. From our proof we know that the map δ is a principal T -bundle and T permutes transitively the leaves lying over Cw We thus have a canonical stratification of B` , indexes by the Weyl group, by the subsets xw . Each such subset is a union of leaves permuted transitively by the right multiplications of the group T . We say that a point a P Spec Z0` “ B` lies over w if ψpaq P Cw . _

4.4 Recall that T “ Zˆ bZ A_ and therefore any λ P P “ HomZ pQ , Zˆ q defines a homomorphism (again denoted by)λ : T Ñ Zˆ For each t P T we define and automorphism βt of the algebra BE by: βt pKα q “ αptqKα ,

βt pEα q “ αptqEα .

Note that the automorphisma Bt leave Z0` invariant and permute transitively the leaves of each set ψ´1 pCw q Ă B` . Given a P B` “ Spec Z0` , denote by ma the corresponding maximal ideal of Z0` and let Aa “ BE {ma BE . These are finite-dimensional algebras and we may also consider these algebras as algebras with trace in order to use the techniques of [DKP2]. 62

Theorem. If a, b P Spec Z0` lie over the same element w P W, then the algebras Aa and Ab are isomorphic (as algebras with trace). Proof. We just apply Proposition 4.1 to the vector bundle of algebras Aa over a symplectic leat and the group T of algebra automorphisms which permutes the leaves in ψ´1 pCw q transitively. 4.5 Let Bw :“ B` X wB´ w´1 and U w :“ U` X wU´ w´1 so that Bw “ U w T . Set also Uw :“ U` X wU` w´1 . One knows that dim Bw “ n ` lpwq and that the multiplication map: σ : Uw ˆ Bw Ñ B` 72

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is an isomorphism of algebraic varieties. We define the map pw : B` Ñ Bw to be the inverse of σ followed by the projection on the second factor. Proposition. The map pw |xw : Xw Ñ Bw is birational. Proof. We need to exhibit a Zarisko open set Ω Ă Bw such that for any b P Ω there is a unique u P Uw with ub P xw . Let n P NG pT q be as above a representative for w so that: Xw “ tb P B` |b “ b1 nb2 , where b1 , b2 P B´ u. Consider the Bruhat cell B´ n´1 B´ Ă G. Every element in B´ n´1 B´ can be written uniquely in the form: bn´1 u, where b P n´1 Bw n, u P U´ . The set B` U´ “ B` B´ is open dense and so it intersects B´ n´1 B´ “ n´1 bw U´ in a non-empty open set which is clearly B´ -stable for the right multiplication, hence B` B´ X B´ n´1 B´ “ n´1 ΩU´ for some non empty open set Ω Ă Bw . In particular Ω Ă nB` B´ “ nUw Bw U´ . Take b P Ω and write it as b “ nxcv with x P Uw , c P Bw , v P U´ . By the remarks made above this decomposition is unique; furthermore, nxn´1 P Uw , ncn´1 P B´ . For the element n´1 b “ `xcv we˘ have 63 by construction that xcv P B´ n´1 B´ and nx´1 n´1 b “ ncn´1 nv P B´ nB´ and nx´1 n´1 P Uw . Thus setting u :“ nx´1 n´1 we have found u P Uw such that ub P Xw . This u is unique since the element x is unique. We are ready now for the concluding theorem which is in the spirit of the conjecture formulated in [DKP1]. 73

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Theorem. Let p P Xw be a point over w P W and let A p be the corresponding algebra. Assume that l is good integer. Then the dimension of 1 each irreducible representation of A p is divisible by =l 2 plpwq`rankp1´wqq . Proof. Consider the algebra BwE for which we know by Theorem 3.5 that 1

deg Bw “ l 2 plpwq`rankp1´wqq . The subalgebra Z0,w of Z0 generated by the elements Kλl and Eαl , where λ P P and α P R` is such that ´w´1 α P R` , is isomorphic to the coordinate ring of Bw , and BwE is a finite free module over Z0,w . Thus by [DKP2] there is a non empty open set A of Bw such that for p P A any irreducible representation of Bw lying over p is of maximal dimension, equal to the degree of BwE . Now the ideal I defining Xw has intersection 0 with Z0,w and so when we restrict a generic representation of BE laying over points of Xw to the algebra BwE we have, as a central character of Z0,w , a point in A Thus the irreducible representation restricted to Bw has all its composition factors irreducible of dimension equal to deg Bw . This proves the claim. It is possible that the dimension of any irreducible representation of BE whose central character restricted to Z0` is a point of xw is ex1 actly ℓ 2 pℓpwq`rankp1´wqq . This fact if true would require a more detailed analysis in the spirit of Section 1.3. We would like, in conclusion, to propose a more general conjecture, similar to one of the results of [WK] on solvable Lie algebras of characteristic p. ‰ “ Let A be an algebra over IC q, q´1 on generators x1 , . . . , xn satisfying the following relations: xi x j “ qhi j x j xi ` Pi j if i ą j,

“ ‰ where phi j q is a skew-symmetric matrix over Z and Pi j P IC q, q´1 rx1 , . . . , xn s. Let ℓ ą 1 be an integer relatively prime to all elementary divisors of the matrix phi j q and let Aε ““ A{pq ´ εq‰ and assume that all elements xiℓ are central. Let Z0 “ IC x1ℓ , . . . , xnℓ ; this algebra has a canonical Poisson structure. 74

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Conjecture. Let π be an irreducible representation of the algebra Aε and 64 let Oπ Ă Spec Z0 be the symplectic leaf containing the restriction of the central character of π to Z0 . Then the dimension of this representation 1 is equal to ℓ 2 dim Oπ . This conjecture of course holds if all Pi j are 0, and it is in complete agreement with Theorems 1.6, 3.5 and 4.6.

References [C] R.W. Carter, Conjugacu classes in the Weyl group, Compositio Math. 25 (1972) 1-59. [D] V.G. Drinfeld, Quantum groups, Proc.ICM Berkeley 1 (1986) 789820. [DK1] C. De Concini, V.G. Kac, Representations of quantum groups at roots of 1, in progress in Math. 92, Birksh¨auser, (1990) 471-506. [DK2] C. De Concini, V.G. Kac, Representations of quantum groups at roots of 1 : reduction to the exceptional case, in Infinite Analysis, World Scientific, (1992) 141-150. [DKP1] C. De Concini, V.G. Kac, C. Procesi, Quantum coadjoint action, Journal of AMS, 5 (1992) 151-190. [DKP2] C. De Concini, V.G. Kac, C. Procesi, Some remarkable degenerations of quantum groups, preprint, (1991). [GL] K.R. Goodearl, E.S. Letzter, Prime ideals in skew and q-skew polynomial rings, preprint (1991). [KP] V.G.KAc, D.H. Peterson, Generalized invariants of groups generated by reflections, in Progress in Math. 60, Birkh¨auser, (1985) 231-250. [L] G. Lusztig, Quantum groups at roots of 1, Geom. Ded. 35 (1990) 89- 114. 75

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[LS] S.Z. Levendorskii, Ya. S. Soibelman, Algebras of functions on compact quatum groups, Schubert cells and quantum tori, Comm. Math. Physics 139 (1991) 141-170. [LW] J.-H. Lu, A Weinstein, Poisson Lie groups, dressing transformations and Bruhat decompositions, J. Diff. Geom 31 (1990) 501526. [STS] M.A. Semenov-Tian-Shansky, Dressing transformations and Poisson group actions, Publ. RIMS 21 (1985) 1237-1260. 65

[WK] B. Yu, Weisfeiler, V.G. Kac, On irreducible representations of Lie p-algebras, Funct. Anal. Appl. 5:2 (1971) 28-36. Classe di Scienze Scuda Normale Superiore 56100 Pisa Piazza dei Cavalieri Italy

Department di mathematics Instituto Guide Gastelnuovo Univrsita di Ronala Sapienza Piazzale Aldo Moro, 5 1-00185 Rome, Italy

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Compact complex manifolds whose tangent bundles satisfy numerical effectivity properties Jean-Pierre Demailly (joint work with Thomas Peternell and Michael Schneider) Dedicated to M. S. Narasimhan and C.S. Seshadri on their sixtith birthdays

0 Introduction A compact Riemann surface always ha s hermitian metric with constant 67 curvature, in particular the curvature sign can be taken to be constant: the negative sign corresponds to curves of general type (genus ě 2), while the case to zero curvature corresponds to elliptic curves (genus 1), positive curvature being obtained only for P1 (genus 0). In higher dimensions the situation is must more subtle and it has been a long standing conjecture due to Frankel to characterize Pn as the only compact K¨ahler manifold with positive holomorphic bisectional curvature. Hratshorne strengthened Frankel’s conjecture and asserted that Pn is the only compact complex manifold with ample tangent bundle. In his famous paper [Mo79], Mori solved Hartshorne’s conjecture by using characteristic p methods. Around the same time Siu and Yau [SY80] gave an analytic proof of the Frankel conjecture. Combining algebraic and analytic tools Mok [Mk88] classfied all compact K¨ahler manifolds with semi-positive holomorphic bisectional curvature. From the point of view of algebraic geometry, it is natural to consider the class fo projective manifolds X whose tangent bundle in numerically effective (nef). This has been done by Campana and Peternell [CP91] and - in case of dimension 3 -by Zheng [Zh90]. In particular, a complete classification is obtained for dimension at most three. The main purpose of this work is to investigate compact (most often K¨ahler) manifolds with nef tangent or anticanonical bundles in arbitrary 77

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dimension. We fist discuss some basic properties of nef vector bundles which will be needed in the sequel in the general context of compact complex manifolds. We refer to [DPS91] and [DPS92] for detailed proofs. Instead, we put here the emphasis on some unsolved questions.

1 Numerically effective vector bundles In algebraic geometry a powerful and flexible notion of semi-positivity is numerical effectivity(“nefness”). We will explain here how to extend this notion to arbitrary compact complex manifolds. Definition 1.1. A line bundle L on a projective manifold X is said to be numerically effective (nef for short) if L ¨ C ě 0 for all compact curves C Ă X. It is cleat that a line bundle with semi-positive curvature is nef. The converse had been conjectured by Fujita [Fu83]. Unfortunately this is not true; a simple counterexample can be obtained as follows: Example 1.2. Let Γ be an elliptic curve and let E be a rank 2 vector bundle over Γ which is a non-split extension of O by O; such a bundle E can be described as the locally constant vector bundle over Γ whose monodromy is given by the matrices ˆ ˙ ˆ ˙ 1 0 1 1 , 0 1 0 1 associated to a pair of generators of π1 pΓq. We take L “ OE p1q over the ruled surface X “ PpEq. Then L is nef and it can be checked that, up to a positive constant factor, there is only one (possibly singular) hermitian metric on L with semi-positive curvature; this metric is unfortunately singular and has logarithmic poles along a curve. Thus L cannot be semi-positive for any smooth hermitian metric. Definition 1.3. A vector bundle E is called nef if the line bundle OE p1q is nef on PpEq (= projectivized bundle of hyperplanes in the fibres of E). 78

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Again it is clear that vector bundle E which admits a metric with semi-positive curvature (in the sense of Griffiths) is nef. A compact K¨ahler manifold X having semi-positive holomorphic bisectional curvature has bu definition a tangent bundle T X with semi-positive curvature. Again the converse does not hold. One difficulty in carrying over the algebraic definition of nefness to the Kahler case is the possible lack of curves. This is overcome by the following: Definition 1.4. Let X be a compact complex manifold with a fixes her- 69 maitian metric ω. A line bundle L over X in nef if for every ε ą 0 there exists a smooth hermitian metric hε on L such that the curvature satisfies Θhε ě ´εω. This means that the curvature of L can have an arbitrarilly small negative part. Clearly a nef line bundle L satisfies L ¨ C ě 0 for all curves C Ă X, but the coverse in not true (X may have no curves at all, as is the case for instance for generic complex tori). For projective algebraic X both notions coincide; this is an easy consequence of Seshadri’s ampleness criterion: take L to be a nef line bundle in the sense of Definition 1.1 and let A be an ample line bundle; then LbK b A is ample for every integer k and thus L has smooth hermitian metric with curvature form ΘpLq ě ´ 1k ΘpAq. Definition 1.3 can still be used to define the notion of nef vector bundles over arbitrary compact manifolds. If pE, hq is a hermitian vector bundle recall that the Chern curvature tensor ÿ i 2 DE,h “ i a jkλµ dz j ^ dzk b e‹λ b eµ Θh pEq “ 2π 1ď j,kďn 1ďλ,µďr

is a hermitian (1,1)-form with values in HompE, Eq. We say that pE, hq is semi-positive inřGriffiths’ sence [Gr69] and write Θh pEq ě 0 if Θh pEqpξ b tq “ a jkλµ ξ j ξk vλ vµ ě 0 for every ξ P T x X, v P E x , x P X. We write Θh pEq ą 0 in case there is strict inequality for ξ ‰ 0, mv ‰ 0. Numerical effectivity can then be characterized by the following differential geometric criterion (see [De91]). 79

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Criterion 1.5. Let ω be a fixed hermitian metric on X. A vector bundle E on X is nef if and only if there is a sequence of hermitian metrics hm on S m E and a sequence εm of positive numbers decreasing to 0 such that Θhm pS m Eq ě ´mεm ω b IdS m E in the sense of Griffiths. The main functional properties of nef vector bundles are summarized in the following proposition. Proposition 1.6. Let X be an arbitrary compact complex manifold and let E be a holomorphic vector bundle over X. 70

(i)

If f : Y Ñ X is a holomorphic map with equidimensional fibres, then E is nef if and only if f ‹ E is nef. *

(ii) Let Γa E be the irreducible tensor representation of GlpEq of highest weight a “ pa1 , . . . ar q P Zr , with a1 ě . . . ě ar ě 0, Then Γa E is nef. In particular, all symmetric and exterior powers of E are nef. (iii) let F be a holomorphic vector bundle over x. If E and F are nef, then E b F is nef. (iv) If some symmetric power S m E is nef pm ą 0q, then E in nef. (v) Let 0 Ñ F Ñ E Ñ Q Ñ 0 be an exact sequence of holomorphic vector bundles over X. Then (αq E nef ñ Q nef.

(βq F, Q nef ñ E nef.

(γq E nef, pdet Qq´1 nef ñ F nef. The proof of these properties in the general analytic context can be easily obtained by curvature computations. The argumentsa are parallel * We

expect (70) to hold whenever f is surjective, but there are serious technical difficulties to overcome in the nonalgebraic case.

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to those of the algebraic case and will therefore be omitted (see [Ha66] and [CP91] for that case). Another useful result which will be used over and over in the sequel is Proposition 1.7. Let E be a nef vector bundle over a connected compact n-fold X let σ P H 0 pX, E ‹ q be a non zero section. Then σ does not vanish anywhere. Proof. We merely observe that if hm is a sequence of hermitian metrics in S m E as in criterion 5, then Tm “

i 1 BB log ||σm ||hm π m

has zero BB-cohomology class and satisfies T m ě ´εm ω. It follows that T m converges to a weak limit T ě 0 with zero cohomology class. Thus T “ iBBϕ for some global plurisubharamonic function ϕ on X. By the maximum principle this implies T “ 0. However, if σ vanishes at some point x, then all T m have Lelong number ě 1 at x. Therefore so has T , contradiction. one of out key results is a characterizations of vector bumdles E which are numerically flat, i.e. such that both E and E ‹ are nef. Theorem 1.8. Suppose that X is K¨ahler. Then a holomorphic vector 71 bundle E over X in numerically flat iff E admits a filtration t0u “ E0 Ă E1 Ă . . . Ă E p “ E by vector subbundles such that the quotients Ek {Ek´1 are hermitian flat, i.e. given by unitary representations π1 pXq Ñ Uprk q. Sketch of Proof . It is clear by 1.6 (v) that every vector bundle which os filtrated with hermitian flat quotients is nef as well as its dual. Conversely, suppose that E is numerically flat. This assumption implies c1 pEq “ 0 Fix a K¨ahler metric ω. If E is ω-stable, then E is HermiteEinstein by the Unlenbeck-Yau theorem [UY86], Moreover we have 0 ď c2 pEq ď c1 pEq2 by Theorem 1.9 below, so c2 pEq “ 0. Kobayashi’s 81

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flatness that E is hermitian flat. Now suppose that E is unstable and take F Ă OpEq to be destabilizing subsheaf of minimal rank p. We then have by definition c1 pF q “ c1 pdet F q “ 0 and the morphism det F Ñ Λ p E cannot have any zero curvature current on the line bundle det F , contradiction). This implies easily that F is locally free, and we infer that F is also numerically flat. Since F is stable by definition, F must be hermitian flat. We set E1 “ F , observe that E 1 ´ E{E1 is again numerically flat and proceed by induction on the rank. Another key point, which has been indeed used in the above proof, is the fact that the Fulton-Lazarsfeld inequalities [FL83] for Chern classes of ample vector bundles still hold for nef vector bundles over compact K¨ahler manifolds: Theorem 1.9. Let pX, ωq be a compact K¨ahler manifold and let E be a nef vector bundle on x. Then for all positive polynomials p theŹ cohomolş ogy class PpcpEqq is numerically positive, that is, Y PpcpEqq ωk ě 0 for anu k and any subvariety Y of X. By a positive polynomial in the Chern classes, we mean as usual a homogeneous weighted polynomial Ppc1 m . . . , cr q with deg ci “ 2i, such that P is a positive integral combination of Schur polynomials: Pa pcq “ detpcai ´i` j q1ďi, jďr ,

72

r ě a1 ě a2 ě . . . ě ar ě 0

(by convention C0 “ 1 ana ci “ 0 if i ‰ r0, rs, r “ rank E). The proof of Theorem 1.9 is based essentially on the same artuments as the original proof of [FL83] for the ample case: the starting point is the nonnegativity of all Chern classes ck pEq (Bloch-Gieseker [BG71]); the general case then follows from a formula of Schubert calculus known as the Kempf-Laksov formula [KL74], which express any Schur ploynomial Pa pcpEqq as a Chern class ck pFa q of some related vector bundle Fa . The only change occurs in the proof of Gieseker’s result, where the Hard Lefschetz theorem is needed for arbitrary K¨ahler metrics instead of hyperplane sections (fortunately enough, the technique then gets simlified, covering tricks being eliminated). Since c1 ck´1 ´ ck 0 ď ck pEq ď c1 pEqk for all k 82

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Therefore all Chern monomials are bounded above by corresponding powers c1 pEqk of the same degree, and we infer: Corollary 1.10. If E in nef and c1 pEqn “ 0, n “ dim X, then all Chern polynomials PpcpEqq of degree 2n vanish.

2 Compact K¨ahler manifolds with nef anti-canonical line bundle Compact K¨ahler manifolds with zero or semi-positive Ricci curvature have been investigated by various authors (cf. [Ca57], [Ko61], [Li67], [Li71], [Li72], [Bo74a], [Bo74b], [?], [Ko81] and [Kr86]). The purpose of this section is to discuss the following two conjectures. Conjecture 2.1. Let X be a compact K¨ahler manifold with numerically effective anticanonical bundle KX´1 . Then the fundametal group π1 pXq has polynomial growth. Conjecture 2.2. Let x be a compact K¨ahler manifold with KX´1 numerically effective. Then the Albanese map α : X Ñ AlbpXq is a smooth fibration onto the Albanese torus. If this hold, one can infer that there r has simply connected fibres. In particular, πi pXq is a finite e´ tale cover X would almost abelian (namely an extension of a finite group by a free abelian group). These conjectures are known to be true if KX´1 is semi-positive. In both cases, the proof is based in differential geometric techniques (see e.g. [Bi63], [HK78] for Conjecture 2.1 and [Li71] for Conjecture 2.2). However, the methods of proof are not so easy to carry over to the nef case. Our main contributions to these conjectures are derived from Theorem 2.3 below. Theorem 2.3. Let X be a compact K¨ahler manifold with KX´1 nef. Then 73 π1 pXq is a group of subexponential growth. The proof actually gives the following additional fact (this was already known before, see [Bi63]). 83

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Corollary 2.4. If morever ´KX is hermitian semi-positive, then π1 pXq has polynomial growth of degree ď 2 dim X, in particular h1 pX, OX q ď dim X. As noticed by F. Campana (private communication), Theorem 2.3 also implies the following consequences. Corollary 2.5. Let X be a compact K¨ahler manifold with KX´1 nef. Let α : X Ñ AlbpXq be the Albanese map and set n “ dim X, d “ dim αpXq. If d “ 0, 1 od n, α is surjective. The same is true if d “ n ´ 1 and if X is projective algebraic. Corollary 2.6. Let x be a K¨ahler surface or a projective 3-fold with KX´1 nef. Then the Albanese map α : X Ñ AlbpXq is surjective. We now explain the main ideas required in the proof of Theorem 2.3. If G is a finitely generated group with generators g1 , . . . , g p , we denote by Npkq the number of elements γ P G which can be written as words γ “ gεi11 . . . gεikk , ε j “ 0, 1 or ´ 1

of length ď k in terms of the generators. The group G is said to have subexponential growth if for every ε ą 0 there is a constant Cpεq such that Npkq ď Cpεqeεk for k ě 0.

This notion is independent of the choice of generators. In the free group with two generators, we have Npkq “ 1 ` 4p1 ` 3 ` 32 ` ¨ ¨ ¨ ` 3k´1 q “ 2 ¨ 3k ´ 1, thus a group with subexponential growth cannot contain a non abelian free subgroup. The first step consists in the construction of suitable K¨ahler metric on X. Since KX´1 in nef, for every ε ą 0 there exists a smooth hermitian metric hε on KX´1 such that uε “ Θhε pKX´1 q ě ´εω. By [Y77] and [Y78] there exists a unique k¨ahler metric ωε in the cohomology class ω such that Riccipωε q “ ´εωε ` εω ` uε . 84

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In fact uε belongs to the Ricci class c1 pKX´1 q “ c1 pXq, hence so does 74 the right hand side ´εωε ` εω ` uε . In particular there exists a function fε such that uε “ Riccipωq ` iBB fε . If we set ωε “ ω ` iBB fϕ (where ϕ depends on ε), equation (+) is equivalent to the Monge-Amp`ere equation ´ ¯n ω ` iBB “ eεϕ´ fε (++) ωn because iBB logpω ` iBBϕqn {ωn “ Riccipωq ´ Riccipωε q

“ εpωε ´ ωq ` Riccipωq ´ uε “ iBBpεϕ ´ fε q.

It follows from the general results of [Y78] that (++) has a unique solution ϕ, thanks to the fact the right hand side of (++) is increasing in ϕ. Since uε ě ´εω, equation (+) implies in particular that Riccipωε q ě ´εω. Now, recall the well-known differential geometric technique for bounding NpKq (this technique has been explained to us in a very efficient way by S.Gallot). Let pM, gq be a compact Riemannian m-fold r be a fundamental domain for the action of π1 pMq on the and let E Ă M r Fix a P E and set β ´ diam E . Since π1 pMq acts universal covering M. r isometrically on M with respect to the pull-back metric g, we infer that ď Ek “ γpEq γPπ1 pMq, lengthpγqďk

has volume equal to Npkq VolpMq and is contained in the geodesic ball Bpa, αk ` βq, where α is maximum of the length of loops representing the generators g j . Therefore NpKq ď

VolpBpa, αk ` βqq VolpMq 85

(˚)

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r For this and it is enough to bound the volume of geodesic balls in M. we use the following fundamental inequality due to R. Bishop [Bi63], Heintze-karcher [HK78] and M. Gage [Ga80]. Lemma 2.7. Let

75

r Ñ M, r Φ : Ta M

Φpζq “ expa pζq

be the (geodesic) exponential map. Denote by Φ˚ dVg “ apt, ζq dt dσpζq the exrpression of the volume element in spherical coordinates with t P r Suppose that apt, ζq does R` and ζ P S a p1q “ unit spheren in T a M. not vanish for t Ps0, τpζqr, wiht τpζq “ `8 or apτpζq, ζq “ 0 Then bpt, ζq ´ apt, ζq1{pm´1q satisfies on s0, τpζqr the inequality B2 1 Riccig pvpt, ζq, vpt, ζqqbpt, ζq ď 0 bpt, ζq ` 2 m´1 Bt where vpt, ζq “

d r expa ptζq P S Φptζq p1q Ă T Φptζq M. dt

If Riccig ě ´εg, we infer in particular

ε B2 b ´ bď0 2 m´1 Bt

a ε{pm ´ 1q (to check and therefore bpt, ζq ď α´1 sinhpαtq wiht α “ this observe that bpt, ζq “ t ` optqq at 0 and that sinhpαtqBb{Bt ´ α coshpαtqb has a negtive derivative). Now, every point x P Bpa, rq can be joined to a4 by a minimal geodesic art of lenght ă r. Such a geodesic are cannot contain any focal point (i.e. any critical value of Φ), except possibly at the end point x. It follows that Bpa, rq is the image be Φ of the open set Ωprq “ tpt.ζq P r0, rrˆS a p1q; t ă τpζqu. 86

87

Compact complex manifolds whose tangent bundles... Therefore Volg pBpa, rqq ď

ż

Ωprq

Φ˚ dVg “

ż

bpt, ζqm´1 dt dσpζq.

Ωprq

As α´1 sinhpαtq ď teαt , we get żr ż ? dσpζq tm´1 epm´1qαt dt ď vm rm e pm´1qεr Volg pBpa, rqq ď S a p1q

0

(˚˚)

Rm .

where vm is the volume of the unit ball in In our application, the difficulty is that the matrix g “aωε varies with ε as well as the constants α “ αε, β “ βε in (˚), and αε pm ´ 1qε need nit converge to 0 as ε tents to 0. We overcome theis difficulty by the following lemma. r Then for every δ ą 0, 76 Lemma 2.8. Let U1 , U2 be compact subsets of X. there are closed subsets U1,ε,δ Ă U1 and U2,ε,δ Ă U2 with Volω pU j U j,ε,δ q ă δ, such that any two points x1 P U1,ε,δ , x2 P U3,ε,δ can be joined by a path of length ď Cδ´1{2 with respect to ωε , where C is a constant independent of ε and δ. We will not explain the details.The basic observation is that ż ż ωε ^ ωn´1 “ ωn X

x

does not depend on ε, therefore ||ωε ||L1 pXq is uniformly bounded. This is enough to imply the existence of suffciently many paths of bounded lenght between random points taken in X (this is done for example by computing the average lenght of piecewise linear paths). We let U be a comnpact set containing the fundamental domain E, so large that U ˝ X gj pU ˝ q ‰ H for each generator g j . We apply Lemma 2.8 with U1 “ U2 “ U and δ ą 0 fixed such that δă

1 Volω pEq, 2

δă

1 Volω pU X g j pUqq. 2

We get Uε,δ Ă U with Volω pUUω,δ q ă δ and diamωε ď Cδ´1{2 The inequalities on volumes imply that Volω pUε,δ X Eq ě 21 Volε pEq and 87

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Uε,δ X g j pUε,δ q ‰ H for every j (note that all g j preserve volumes). It is then clear that ď Wk,ε,,δ :“ γpUε,δ q γPπ1 pXq,lengthpγqďk

satisfies 1 Volω pWk,ε,δ q ě Npkq Volω pUε,δ X Eq ě Npkq Volω pEq 2 diamωε pWk,ε,δ q ď k diamωε Uε,δ ď kCδ´1{2 .

and

Since m “ dimR X “ 2n, inequality (˚˚) implies Volωε pWk,ε,δ q ď Volωε pBpa, kCδ´1{2 qq ď C4 k2n eC5

? εk

.

Now X is compact, so there is a constant Cpεq ą 0 such that ωn ď Cpεqωnε . We conclude that NpKq ď

2 Volω pWk,ε,δ q ? ď C6Cpεqk2n eC5 εk . Volω pEq

The proof of Theorem 2.3 is complete. 77

Remark 2.9. It is well known and easy to check that equation (++) implies ˙ ˆ Cpεq ď exp max fε ´ min fε . X

X

Therefore it is reasonable to expect the Cpεq has polynomial growth in ε´1 ; this would imply that π1 pXq has polynomial growth by taking ε “ k´2 . When KX´1 has a semipositive metric, we can even take ε “ 0 and find a metric ω0 with Riccipω0 q “ u0 ď 0. This implies Corollary 2.4. Proof of Corollary 2.5. If d “ 0, then by definition H 0 pX, Ω1X q “ 0 and AlbpXq “ 0. If d “ n, the albanese map has generic rank n, so there exist holomorphic 1-forms u1 , . . . , un such that u1 ^ . . . ^ un ı 0. How ever 88

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89

u1 ^ . . . ^ un is a section of KX which has a nef dual, so u1 ^ . . . ^ un cannot vanish by Proposition 1.7 and KX is trivial. Therefore u1 ^ . . . uˆ k . . . ^ un ^ v must be a constant for every holomorphic !-form v and pu1 , ldots, un q is a basis of H ˝ pX, Ω1X q. This implies dim ApXq “ n, hence α is surjective. If d “ 1, the image C “ αpXq is a smooth curve. The genus g of C cannot be ě 2, otherwise π1 pXq would be mapped onto a subgroup of finite index in pi1 pCq, and thus would be of exponential growth, contradicting Theorem 2.3 Therefore C is an elliptic curve and is a subtorus of AlbpXq. By the universal property of the Albanese map, this is possible only if C “ AlbpXq. The case d “ n ´ 1 is more subtle and uses Mori theory (this is why we have to assume X projective algebraic). We refer to [DPS92] for the details.

3 Compact complex manifolds with nef tangent bundles Several interesting classes of such manifolds are produces by the following simple observation. Proposition 3.1. Every homogeneous compact complex manifold has a nef tangent bundle. Indeed, if X is homogeneous, the Killing vector fields generate T X, so T X is a quotient of a quotient of a trivial vector bundle. In praticular, we get the following Examples 3.2. (homogeneous case) (i) Rational homogeneous manifolds: Pn , flag manifolds,quadrics Qn (all are Fano manifolds, i.e. projective algebraic with KX´1 ample.) (ii) Tori IC {Λ (K¨ahler, possibly non algebraic). (iii) Hopf manifolds IC 0{H where H is a discrete group of homotheties (non K¨ahler for n ě 2). 89

78

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Jean-Pierre Demailly

(iv) Iwasawa manifolds G{Λ where G is the group of unipotent upper triangular p ˆ p matrices and λ the subgroup of matrices with entries in the ring of integers of some imaginary quadratic field. eg. Zris (non K¨ahler for p ě, although T X is trivial). We must remark at this point that not all manifolds X with nef tangent bundles are homogeneous, the automorphism group may even be discrete: Example 3.3. Let Γ “ IC {pZ ` Zτq, Imτ ą 0, be ana elliptic curve. Consider the quotient space X “ pΓˆΓΓq{G where G “ 1, g1 , g2 , g1 g2 » Z2 ˆ Z2 is given by ˙ ˆ 1 g1 pz1 , z2 , z3 q “ z1 ` , ´z2 , ´z3 , 2 ˆ ˙ 1 1 g1 pz1 , z2 , z3 q “ ´z1 , z2 ` , ´z3 ` , 2 2 ˙ ˆ 1 1 1 . g1 g2 pz1 , z2 , z3 q “ ´z1 ` , ´z2 ` , z3 ` 2 2 2 Then G acts freely, so X is smooth. It is clear also that T X is nef (in fact T X is unityu flat). Since the pull-back of T X to Γ ˆ Γ ˆ Γ is trivial, we easily conclude that T X has no sections, thanks to the change of signs in g1 , g2 , g1 g2 . Therefore the automorphism group AutpXq is discrete. The same argument shows that H 0 pX, Ω1x q “ 0. Example 3.4. Let X be the ruled surface bbPpEq over the elliptic curve Γ “ ICpZ ` Zτq defined in Example 1.2. Then the relative tangent bundle of bbPpEq Ñ Γ(=relative anticanonical line bundle) is π‹ pdet E ‹ q b OE p2q » OE p2q and T Γ is trivial, so T X is nef. Moreover X is almost homogeneous, with automorphisms induced by px1 , z1 , z2 q ÞÑ px ` a, z1 ` b, z2 q,

pa, bq P IC2

and a single closed orbit equal to the curve tz2 “ 0u. Here, no finite e´ tale cover of X can be homogeneous, otherwise KX´1 “E p∈q would be 90

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91

semi-positive. Observe that no power of KX´1 is generated by sections, although KX´1 in nef. Our main result is structure theorem on the Albanese map of compact K¨ahler manifolds with nef tangent bundles. Main Theorem 3.5. Let X be a compact K¨ahler manifold with nef tan- 79 r be a finite e´ tale cover of X of maximum irregugent bundle T X. Let X, 1 r r O r q. Then larity q “ qpXq “ h pX, X r » Z 2q . (i) π1 pXq

r Ñ ApXq r is a smooth fibration over a (ii) The albanese map α : X q-dimensional torus with nef relative tangent bundle.

(iii) The fibres F of α are Fano manifolds with nef tangent bundles.

Recall that a Fano manifold is by definition a compact comples manifold with ample anticanonical bundle KX´1 . It is well known that Fano manifolds are always simply connected (Kobayashi [Ko61]). As a consequence we get Corollary 3.6. With the assumtions of 3.5. the fundamental group π1 pXq is an extension of a finite group by Z 2q . In order to complete the classification of compact K¨ahler maniflods with nef tangent bundles (up to finite e´ tale cover), a solution of the following two conjectures would be nechap5-enum-(i)eded. Conjecture 3.7. (Campana- Peternell [CP91]) Let X be ab Fano manifold Then X has a nef tangent bundle of and only if X i rational homogeneous. The evidence we have for Conjecture 3.7 is that it is true up to dimension 3. In dimension 3 there are more than 100 different types of Fano manifolds, but only five types have a nef tangent bundle, namely bbP3 , Q3 (quadric), P1 ˆ P2 , P1 ˆ P1 , P1 ˆ P1 and the flag manifold F1,2 of lines and planes in IC3 ;m all five are homogeneous. 91

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A positive solution to Conjecture 3.7 would clarify the structure of fibers in the Albanese map of Theorem 3.5. To get a complete picture of the situation, one still needs to know how the fibers are deformed and glued together to yield a holomorphic family over the Albanese is relatively ample, thus for m large the fibres torus. We note that K ´1 r X can be embedded in the projectivized bundle of the direct image bundle . The structure of the deformation i described by the following α˚ pK ´m r X theorem. Theorem 3.8. In the situation of Theorem 3.5, all direct image bundles Em “ α˚ pK ´m q are numerically flat over the Albanese torus. Moreover, r X for p " m " 0, the fibers of the Albanese map can ne described as Fano submanifolds of the fibers of IPpEm q defined by polynomial equations of degree p, in such a way that the bundle of equations Vm,p Ă S p pEm q is itself numerically flat. 80

Theorem 3.8 is proved in [DPS91] in case X is projective algebraic. The extension to the K¨ahler case has been obtatined by Ch. Mourougane in his PhD Thesis work (Grenoble, still unpublished). We now explain the main steps in the proof of Theorem 3.5 One of the key points is the following Proposition 3.9. Let X be a compact K¨ahler n-fold with T X nef. Then (i) If c1 pXqn ą 0, then X is a Fano manifold. (ii) If c1 pXqn “ 0, then χpOX q “ 0 and there exists a non zero holor Ñ X morphic p-form, p suitable odd and a finite e´ tale cover X r ą 0. such that qpXq

Proof. We first check that every effective divisor D of X in nef. In fact, let σ P H 0 pX, OpDqq be a section with divisor D. Then for k larger than the maximum vanishing order of σ on X, the k-jet section jk σ P H 0 pX, jk OpDqq has no zeroes. Therefore, there is an injection O Ñ J k OpDq and a dual surjection pJ k OpDqq‹ b OpDq Ñ pDq. 92

Compact complex manifolds whose tangent bundles...

93

À Now , J k OpDq has a filtration whose graded bundle is 0ďpďk S p T ‹ Xb OpDq, so pJ k OpDqq‹ b OpDq has a dual filtration with graded bundle À p k ‹ 0ďpďk S T X. By 1.6 (70) and 1.6 (v)(β), we conclude that pJ OpDqq b OpDq is nef, so its quotient OpDq in nef by 1.6 (v) (α). Part (70) is based on the solution of the Grauert-Riemenschneider conjecture as proved in [De85]. Namely, L “ KX´1 “ Λn T X is nef and satisfies c1 pLqn ą 0, so L has Kodaira dimension n (holomorphic More inequalities are needed at that point because X is not suppose a priori to be algebraic). It follows that X is Moishezon,thus projective algebraic, and for m ą 0 large we have Lm “ OpD ` Aq with divisors D, A such that D is effective and A ample. Since D must be in fact nef, it follows that L “ KX´1 is ample, as desired. The most difficult part is (ii). Since c1 pXqn “ 0, Corollary 1.10 imp plies χpOX q “ 0. By Hodge symmetry, we get h0 pX, ΩX q “ h p pX, OX q and ÿ p χpOX q “ p´1q p h0 pX, ΩX q “ 0. 0ďpďn

From this and the fact that h0 pX, OX q “ 1, we infer the existence of a non zero p-form u for some suitable odd number p. Let σ : Λ p´1 T X Ñ Ω1X be the bundle moriphism obtained by contracting pp ´ 1q-vectors with 81 u. For every k ą 0, the morphism Λk σ can be viewed as section of the bundle Λk pΛ p´1 T ‹ Xq b Λk T ‹ X which has nef dual. Hence by Proposition 1.7 we know the Λk σ is either identically zero or does not vanish. This mean is that σ has constant rank. Let E be the image of σ. Then E is a quotient bundle of Λ p´1 T X, so E in nef, and E is subbundle of Ω1X “ T ‹ X, so E ‹ is likewise nef. Theorem 1.8 implies the existence of a hermintian flat subbundle E1 Ă E. If E1 would be trivial after pullingr we would get a trivial subbundle of back to some finite etale cover X, 1 r Ω r , hence qpXq ą 0 and the proposition would be proved. Otherwise X E1 is given by some infinite representation of π1 pXq inti some unitary group. Let Γ be the monodromy group (i.e. the image of π1 pXq by the representation). We use a result of Tits [Ti72] according to which every 93

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subgroup contains either a non abelian free subgroup or a solvable subgroup of finite index. The first case cannot occur by Theorem ??.?. In the second case,we may assume Γ solvable by taking some finite e´ tale cover. We consider the series of derived groups Γ Ą Γ1 Ą . . . Ą Γ N “ 0 and the largest index k such that Γk has finite index in Γ. Then the r of X with inverse image of Γk in π1 pXq defines a finite e´ tale cover X infinite first homologu group (the representation maps this group onto r ą 0, as desired. Γk {Γk`1 which is infinite). Hence qpXq

Proof of the Main Theorem. Let X be compact K¨ahler mainfold with nef tangent bundle. Since a son zero holomorphic form u P H 0 pX, Ω1X q can never vanish by Proposition 1.7, it follows immediately that the Albanese map α has rank to qpXq at very point, hence α is a submersion r be a finite e´ tale cover with maximum irregularity and qpXq ď n. Let pXq r r also a nef tangent bundle, so qpXq r ď n). let q “ qpXq (note that pXq r Ñ ApXq r The relative F denote the fibers of the Albanese map α : pXq tangent bundle exact sequence. dα

0 Ñ T F Ñ T X ÝÑ α‹ T ApXq Ñ 0. in which T ApXq is trivial shows by 1.6 (v)pγq that T F in nef. Lemma r “ 3.10 (iii) below implies that all finite e´ tale covers Fr of F satisfy qpFq O. Hence the fibers F must be Fano by proposition 3.9 and the main Theorem follows.

82

Lemma 3.10. Let X, Y be compact K¨ahler manifolds and let g : X Ñ Y be a smooth fibration with connected fibers. We let qpXq be the irregularity of X and qrpXq be the sup of the irregularity of all finite e´ tale covers. If F denotes any fibre of g, then (i) qpXq ď qpYq ` qpFq,

(ii) qrpXq ď qrpYq ` qrpFq.

94

Compact complex manifolds whose tangent bundles...

95

(iii) Suppose that the boundary map π2 pYq Ñ π1 pFq is zero, that π1 pFq contains an abelian subgroup of finite indes and that Y contains a subvariety S with π1 pS q » π1 pYq, such that any two generic points in the universal covering Sˆ can be joined thorugh a chain of holomorphic images IC Ñ Sˆ . Then qrpXq “ qrpYq ` qrpFq.

The proof is based on a use of the Leray spectral sequence and a study of the resultig monodromy on H 1 pF, ICq. Triviality of the monodromy is achieved in case (iii) becase all K¨ahler deformations of tori over Y must be trivial. We refer the reader to [DPS91] for the details. In our application, Y is taken to be the Albanese torus, so assumption (iii) is satisfied with S “ Y (π1 pFq contains an abelian subgroup of finite indes thanks to Corollary 3.6, by using an induction on dimension).

4 Classification in dimension 2 and 3 By using the Kodaira classification of surface and the structure theorems of Section 3, it is not difficult to classify all K¨ahler surface with nef tangent bundles; except for tori, the K¨ahler classification in identical to the projective one. The projective case was already mentioned in [CP91] and [Zh90]. Theorem 4.1. Let X be a smooth K¨ahler surface such that T X is nef. Then X is minimal and is exactly one of the surfaces in the following list: (i) X is torus; (ii) X is hyperellipitic; (iii) X “ P2 ; (iv) X “ P1 ˆ P1 ; 95

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(v) X “ PpEq, where E is a rank 2-vector bundle on an elliptic curve C with either [pαq ] E “ O ‘ L, L P Pic0 pCq, or 83

[pβq ] E is given by a non split extension 0 Ñ O Ñ E Ñ L Ñ 0 with L “ O or deg L “ 1.

The list of non-k¨ahler surfaces in the Kodaira classification is much smaller. It is then rather easy to check nefness in each case: Theorem 4.2. The smooth non K¨ahler compact comlex surface with nef tangent bundles are precisely: (i) Kodaira surfaces (that is surfaces of Kodaira dimension 0 with b1 pXqx odd); (ii) Hopf-surfaces (that is, surfaces whose universal cover is IC2 0). A similar classification can be obtained for 3-dimensional compact ¨ Khler manifolds. Theorem 7.1. Let X be a K¨ahler 3-fold. Then T X in nef if and only if X is up to finite e´ tale cover one of the manifolds in the following list: (i) X “ P3 ; (ii) x “ Q3 , the 3-dimensional quadric; (iii) X “ P1 ˆ P2 ; (iv) X “ F1,2 , the flag manifolds of subspaces of IC3 ; (v) X “ P1 ˆ P1 ˆ P1 ; (vi) X “ PpEq, with a numerically flat rank 3-bundle on an elliptic curve C; (vii) X “ PpEqˆC PpFq, with E, F numerically flat rank 2-bundles over an elliptic curve C; 96

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(viii) X “ PpEq, with E a numerically flat rank 2-bundle over a 2dimensional complex torus; (ix) X “ 3-dimensional complex torus. The only non-algebraic manifolds appear in classes (viii) and (ix) when the Albanese torus is not algebraic. Let us mention that the classification of projective 3-flods with nef tangent bundles was already carried out in [CP91] and [Zh90]. In addition to Theorem 3.5, the main ingredient is the classfication of Fano 3-folds by Shokrov and MoriMukai. An insepection of the list yields the first classes (i)-(v)

References [Bi63] Bishop. R, A relation between volume, mean curvature and di- 84 ameter, Amer. Math. Soc. Not. 10 (1963) 364. [BG71] Bloch. S, Gieseker. D, The positivity of the Chern classed of an ample vector bundle, Inv. Math. 12 (1971) 112-117. [Bo74a] Bogomolov. F. A, On the decompsition of K´ahler manifolds with trivial coanonical class, Math. USSR Sbornik 22 (1974) 580583. [Bo74b] Bogomolov. F.A, K¨ahler manifolds with trivial canonical class, Izvestija Akad. Nauk 38 (1974) 11-21 and math. USSR Izvestija 8 (1974) 9-20. [BPV84] Bath. w, Peters. C, Van de ven. A, Compact Complex Surfaces, Erg. der Math. 3. Folge, Band 4, Springer, Berlin (1984) [Ca57] Calabi. E, On K¨ahler manifolds with vanishing canonical class, Alg. geometry and topology, Symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton (1957) 78-89. [CP91] Cmapana. F, Peternell. Th, Projective manifolds whose tangent bundles are numerically effective, Math. Ann. 289 (1991) 169-187. 97

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[De85] Demailly. J. P, Champs mag´etiques et in e´ galiti´es de Morse pour la d”-cohomologie, Ann. Inst. Fourier 35 (1985) 185-229. [De91] Demaily. J. P, Regularization of closed positive currents and intersection theory, Preprint (1991) [DPS91] Demaily, J. P, Peternell. Th, Schneider M., Compact complex manifolds with numerically effective tangent bundles, Preprint (1991) To appear in J. Alg. Geom. [DPS92] Demailly. J. P, Peternell. Th, Schneider. M, K¨ahler manifolds with numerically effective Ricci class, CoMpositio Math. 89 (1993) 217-240. [FL83] Fulton. W, Lazarsfeld. R, Positive polynomials for ample vector bundles, Ann. Math. 118 (1983) 35-60. [Fu83] Fujita. T, Semi-positive line bundles, J. Fac. Sci. Univ. Tokyo. 30 (1983) 353-378. 85

[Ga80] Gage. M.E, Upper bounds for the first eigenvalue of the Laplace- Beltrami operator, Indiana Univ. J. 29 (1980) 897-912. [Gr69] Giffiths. P, Hermitian differential geometry, Chern classes, and positive vector bundles, In: Global Anlysis, Princeton Math. Series 29 (1969) 185-251. [Ha66] Hartshorne. R, Ample vector bundles, Publ. Math. Inst. Hautes Etud. Sci 29 (1966) 319-394. [HK78] Heintze. E, Karcher. H, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Scient. Ec.e 11 (1978) 451-470. [KL74] Kempf. G, Laksov. D, The determinantal formula of Schubert calculus, Acta Math. 132 (974) 153-162. [Ko61] Kobayashi. S, On compact K¨ahler manifolds with positive definite Ricci tensor, Ann. Math 74 (1961) 570-574. 98

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[Ko81] Kobayashi. S, Recent results in complex differential geometry, Jahresber. Deutsche Math. Ver. 83 (1981) 147-158. [Ko87] Kobayashi. S, Differential geometry of complex vector bundles, Princeton Univ. Press (1987) [Kd66] Kodaira. K, On the structure of compact complex surface II, Amer. J. Math. 88 (1966) 682-721. [Kr86] Koll´ar. J, Higher direct images of dualizing sheaves, Ann. Math 123 (1986) 11-42. [Li67] Lichnerowiz. A, Vari´et´es K¨ahl´eriennes et premi`ere classe de Chern, J. Diff. Geom. 1 (1967) 195-224. [Li71] Lichnerowicz. A, Vari´et´es K¨ahl´eriennes a` premi`ere classe de Chern non n´egative et vari´et’es riemanniennes a` courbure de Ricci g´en´eralis´ee non n´egative, J. Diff. Geom. 6 (1971) 47-94. [Li72] Lichnerowicz. A, Vari´et´es K¨ahl´eriennes a` premi`ere classe de Chern non n´gative et situation analogue dans le cas riemannien, Ist. Naz. Alta Mat., Rome, Symposia Math., vol.10, Academic Press, New-York (1972) 3-18. [Mk88] Mok. N, The uniformization theorem for compact K¨ahler man- 86 ifolds of non negative holomorphic bisectional curvature, J. Diff. Geom. 27 (1988) 179-214. [Mo79] Mori. S, Projective manifolds with ample tangent bundles, Ann. Math 110 (1979) 593-606. [My41] Myers. S. B, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941) 401-404. [SY80] Siu Y.T, Yau. S. T, Compact K¨ahler manifolds with positive bisectional curvature, Inv. Math. 59 (1980) 189-204. [Ti72] Tits. J, Free subgroups in linear groups, J. of Algebra 20 (1972) 250-270. 99

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[UY86] Uhlenbeck. K, Yau. S. T, On the existence of Hermitian-YangMills connections in stable vector bundles, Comm. Pure and Appl. Math 39 (1986) 258-293. [Y77] Yau. S.T, Calabi’s Conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA 74 (1977) 1789-1790. [Y78] Yau. S.T. On the Riccie curvature of a complex K¨ahler manifold and the complex Monge-Amp`e re equation I, Comm. Pure and Appl. Math. 31 (1978) 339-411. [Zh90] Zheng. F, On semi-positive threefolds Thesis, Harvard (1990)

Univerit´e de Grenoble Institut Fourier, BP 74 U.R.A. 188 du C.N.R.S. 38402 Saint-Martin d’H`eres, France

100

Universita¨at Bayreuth Mathematisches Institut Postfach 10 12 51 D-8580 Bayreuth, Deutschland

Algebraic Representations of Reductive Groups over Local Fields William J. Haboush*

Introduction 87

This paper is an extended study of the behaviour of simplicial co- sheaves in the buildings associated to algebraic groups, both finite and infinite dimensional. Recently the theory of simplicial sheaves and co-sheaves has found a number of important applications to the representation theory and cohomology theory of finite theory of finite groups (see [T], [RS]), the computations of teh cohomology of arithmetic groups and the problem of admissible representatiosn of P-adic groups and teh Langlands classification ([CW], [BW]). My interest has been, for the most part, the representation theory of semi-simple groups over fields of positive characteristic. In this area, of course, the driving force of much recent work has been the so-called Lusztig characteristic p conjecture [L1] (so called to distinguish it from a number of other equally intersecting Lusztig conjectures). In contemplating this conjecture one is struck by certain resonances with work in admissible representations etc. The line of argument I am hoping to achieve is something like this. One should attempt to use the homoligical algebra of simplicial cosheaves to construct a category of representations of something like the loop group associated to th semisimple group, G, which have computable character theory. Then one should attempt to express the finite dimensional representations of G as virtual representations in the category. Then presumably the “generic decomposition patterns” should be formulae expressing the character of a dual Weyl module in terms of these computable characters. The hope of constructing such a theory has led me to conduct the rather extended exploration below. * This

research was founded in part by the National Science Foundation

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One is immediately tempted to replace harmonic analysis with a purely algebraic theory and to use this theory to do the representationtheoretic computations necessary. My replacement for harmonic analysis is this. Let G be algebraic of simple type over Z. Let K be a field complete with respect to a discrete rank one valuations, let O be the valuation ring and let L be its residue field. Then consider the Bruhat-Tits building of GpKq. It is a simplicial complex. Let G “ GpKq. Let I be its Bruhat-Tits building. Then I is G equivariant. Let k be another field. My idea is to consider the category of G equivariant co-sheaves of k-vector spaces which are, in some sense made precise within, locally algebraic. Then in a manner entirely analogous to the classical notion of rational of rational representative functions, this category admits an injective co-generator. The endomorphism ring of this canonical co-generator is a certain algebra. Let it be denoted H. [T] shows that for suitably finite bG is a certain algebra. Let it be denoted H-module. Then one may sent the class of a finite dimensional G- representation to the alternating sum of the left derived functors of the co-limit of its induced G-co-sheaf in the Grothendieck ring of H. In this context that one would hope to obtain interesting identities relating finite dimensional representation theory to the representation theory of H. I have made certain choices in this discussion. As I am discussing sheaves and co-sheaves on simplicial complexes, I have decided to use the word carapaces for co-sheaves. There are three reasons: the first is that the word, co-sheaves, seems rather cobbled together, the second is that a carapace really would look like a lobster shell or such if one were to draw one and the third is the Leray used the word for sheaves and I don’t like to see such a nice word go to waste. am working for the most part with carapaces rather than sheaves because I am working on an infinite simplicial complex and in moving form limit to co-limit, one in moving from an infinite direct product sort of thing to an infinite co-product sort of thing. Thus,in using limits rather than co-limits, one loses structure just as one does in taking an adic completion of a commutative ring. I have also decided to include a discussion of the homological algebra of carapaces. Now all of this material is some sort of special case of certain kinds of sheaves on sites or the homologi102

Algebraic Representations of Reductive Groups over Local Fields 103 cal algebra of abelian group valued functors, but with all due apologies to those who have worked on those topics, I would prefer to formulate this material in a way which anticipates my intentions. A number of mathematicians have done this sort of thing. Tits and Solomon [S], [T], Stephen Smith and mark Ronan [RS] immediately come to mind. But again, working on a infinite complex has dictated that I reformulate many of the elements for this situation.

1 Carapaces and their Homology 89

Basic references for this section are [Mac] and [Gr]. Basic notions and definition all follow those two sources. A simiplicial complex, X, will be a set verpXq together with a collection of finite subsets of verpXq such that when ever σ P X every subset of σ is in X. These finite subsets of verpXq are the simplices of X. The dimension of σ is its cardinal less one and the dimension of X, if it exists,is the maximum of the dimensions of simplices in X. We shall view X as a category, the morphisms being the inclusions of simlices. We identify verpXq with the zero simplices of X. If X and Y are simplicial complexes a morphism of complexes from X to Y is just a convariant functor from X to Y; a simplicial morphism is a morphism of complexes taking vertices to vertices. Let R be a commutative ring with unit fixed for teh remainder of this work and let ModpRq denote the category of R-modules. Let X be a simplicial complex. Definition 1.1. An R-carapace on X is a convariant functor from X to ModpRq. If A and B are two R-carapaces on X, a morphism of Rcarapaces from A to B is a natural transformation of functors. If A is an R-carapace on X and σ P X is a simplex, then Apσq is called the segment of A along σ. A sequence of morphisms of Rcarapaces will be called exact if and only if the corresponding sequence of segments is exact for each simplex in X. The product (respectively coproduct) of a family of R-carapaces is the R-carapace whose segments and morphisms are the products (respectively co-products) of those in 103

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the family of carapaces. If σ Ď τ is a pair of simplices in X write eτA,σ or eτσ when there is no possibility of confusion for the map from Apσq to Apτq. I will call it the expansion of A form σ to τ. Finally, if A and B are two R-carapaces on X, write HomR,X pA, Bq for the R-module of morphisms of R-carapaces from A to B. Let M be an R-module. Then let MX denote the constant carapace with value M. That is, its segment along any simplex is M and its expansions are all the identity map. In addition for any given simplex, σ, there are two dually defined carapaces, M Òσ and M Óσ defined by: M Òσ pτq “ Mpσ Ď τq

M Òσ pτq “ p0qpσ Ę τq M Óσ pτq “ M σ

90

pσ Ě τq

M Ó pτq “ p0q pσ Ğ τq

(1.2)

(1.3)

In M Òσ , the expansions are the identity for paris τ, γ such that σ Ď τ Ď γ and 0 otherwise. In M Óσ they are the identity for τ, γ such that τ Ď γ Ď σ and 0 otherwise. Lemma 1.4. Let X be a simplicial complex and let M be an R-module and let A be an R-carapace on X. Then, 1. HomR,X pM Òσ , Aq “ HomR pM, Apσqq 2. HomR,X pA, M Óσ q “ HomR pApσq, Mq 3. If M is R-projective, then M Òσ is projective in teh category of Rcarapaces on X. 4. If M is R-injective, then M Óσ is injective in the category of Rcarapaces on X. Proof. Statements (1) and (2) require no proof. Note that the functor which assigns to an R-carapace on X its segment along σ is an exact functor This observation together with (1) and (2) implies (3) and (4). 104

Algebraic Representations of Reductive Groups over Local Fields 105 Proposition 1.5. Let X be a simplicial complex. Then the category of R-carapaces on X has enough projectives and enough injectives. Proof. Let A be any R-carapace on X. We must show that there is a surjective map from a projective R-carapace to X and an injective map from A into an injective R-carapace. For each simplex, σ in X choose a projective R-module, Pσ , with a surjective map πσ mapping Pσ onto Apσq. Let ž P“ Pσ Òσ σPX

For each σ, let πσ be the morphism of carapaces corresponding to πσ given by 1.4, 1. Define π by: ž πσ . π“ σPX

Then π is a surjective map from a projective to A. 91 To construct an injective, choose an injective module and an inclusion, jσ : Apσq Ñ Iσ . Then define an injective carapace, I, and an inclusion, j, as products of the carapaces Iσ Óσ and inclusions jσ defined dually to the corresponding objects in the projective case. Proposition 1.6. Let X be a simplicial complex. Then (1) If P is a projective R-caparapace on X, then Ppσq is R projective for each σ P X. (2) If I is an injective R-carapace on X, then Ipσq is R injective for each σ P X. Proof. Suppose P is projective. For each σ let πσ : Fσ Ñ Ppσq be ašsurjective map from a projective R-module onto Ppσq. Then Q “ š σPX πσ maps Q onto P. Since P σPX F σ Òσ is projective by 1.4 and is projective, it is ašdirect summand of Q. But then Ppσq is a direct summand of Qpσq “ τĎσ Fτ which is clearly projective. This establishes the first statement. To prove the second statement, for each σ choose ś and embedding,, jσ : Ipσq Ñ Jσ where Jσ in R-injective. Then use σPX to embed I in 105

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ś

Jσ and reason dually to the previous argument replacing, at each point where it occurs, the co-product with the product. σPX

It is clear that there are natural homology and cohomology theories on the category of R-carapaces on X. The most natural functors to consider are the limit and the co-limit over X. To simplify the discussion, CarR pXq denote the category of R-carapaces on X. Definition 1.7. Let U denote any sub-category of X, and let A be an R-carapace on X. Then Apσq ΣpU, Aq “ lim ÝÑ σPU

Apσq. ΓpU, Aq “ lim ÐÝ σPU

We will refer to ΣpU, Aq as the segment of A over U and to ΓpU, Aq as the sections of A over U.

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Certain observation are in order. Since subcategories of X are certainly not in general filtering the functor, ΣpU, ?q, is right exact on CarR pXq. Similarly ΓpU, ?q is left exact. Furthermore ΓpU, Aq “ HomR,U pRU , Aq, On the other hand, ΣpU, Aq cannot be represented as a homomorphism functor in any obvious way but its definition as a direct limit allows us to conclude that: HomR pΣpX, Aq, Mq “ HomR,X pA, MX q Definition 1.8. For any R-carapace on X, A, let Hn pX, Aq “ Ln ΣX, A and let H n pX, Aq “ Rn ΓpX, Aq the left and right derived functors of ΣpX, ´q and ΓpX, ´q respectively. These groups shall be referred to as the expskeletal homology and cohomology groups of X with coefficients in A. 106

Algebraic Representations of Reductive Groups over Local Fields 107 Example 1.9. The Koszul Resolution of the Constant Carapace. Choose an ordering on the vertices of X. For each r ě š 0, let Xprq denote the set of simplices of dimension r. Let Kq pX, Rq “ σPXpqq R Òσ . Źq If A is any R-carapace on X,Ź then R A is understood to be the carapace q whose segment along σ is R pApσqq. Then, it is not at all difficult to Źq`1 see that R K0 pX, Rq “ Kq pX, Rq Furthermore, when σ Ď τ there is always a natural map from M Òτ to M Òσ obtained by applying 4,1 to M Òτ and noting that since M Òσ pσq “ M “ M Ò pτq there is a map in HomR,X pM Òτ , MM Òσ q corresponding to the identity. Make use of the ordering on the vertices of X to define an alternating sum of the maps corresponding to the faces of a simplex. It is easy to see the that this gives a complex of carapaces: . . . Ñ pX, Rq Ñ Kq´1 pX, Rq Ñ . . . Ñ K0 pX, Rq Ñ RX Ñ p0q Then check that the sequence of segments on σ is just the standard Koszul resolution of the unit ideal which begina with a free module of rank dimpσq ` 1 and the map which sends each of its generators to one. Consequently, this construction gives a resolution of the constant carapace by projectives. On the other hand it is evident that σpX, Kq pX, Rqq is just the R-module of simplicial q-chains on X with coefficients in R and that the boundaries are the standard simplicial boundaries. In this way, one verifies that the exoskeletal homology and cohomology with coefficients in a constant carapace is just the simplicial homology and cohomology. This phenomenon has been observed and exploited by Casselman and Wigner in their work on admissible representations and the cohomology of artihmetic groups [CW].

2 Operations on Carapaces 93

In the category of R-carapaces on X there is a self-evident notion of tensor product: Definition 2.1. Let A and B be two R-carapaces on X. Let: pA bR Bqpσq “ Apσq bR Bpσq 107

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Then A bR B is a carapace which we will refer to as the tensor product of A and B. The properties of the tensor product are for the most part clear. Most of them are stated in the following. Proposition 2.2. Let X be a simplicial complex. Then the tensor product of R-carapaces on X is an associative, symmetric, bi-additive functor right exact in both variables. Furthermore: (1) For any R-carapace, A, RX bR A » A (2) If P is a projective R-carapace then tensoring with P on either side is an exact functor. (3) For any two R-carapaces, A and B, there is a natural map: tA,B : ΣpX, A bR Bq Ñ ΣpX, Aq bR ΣpX, Bq Moreover, tA,B is a natural transformation in A and B and it is functorial in X as well. Proof. The first statement is self-evident; the second follows trivially from 1.6 but the third might require some comment. To construct tA,B let eA,σ : Apσq Ñ ΣpX, Aq and eB,σ : BpΣq Ñ ΣpX, Bq be the expansions. Then eA,σ bR eB,Σ maps A bR Bpσq into ΣpX, Aq bR ΣpX, Bq compatibly with respect to expansion. Since ΣpX, A bR Bq is a colimit, this defines tA,B uniquely and ensures that it is functorial as asserted. The construction of a tensor product is thus quite straightforward but the construction of an internal homomorphism functor with the requisite adjointness properties presents certain technical difficulties. For any σ P X let xpσq denote the subcategory of X whose objects are the simplices τ such that τ Ě σ. The morphisms of Xpσq are inclusions of simplices. Then Xpσq is not a subcomplex of X. If σ Ď τ then Xpτq Ď Xpσq. There is a natural restriction map from the group HomR,Xpσq pA|Xpσq , B|Xpσq q to HomR,Xpτq pA|Xpτq , B|Xpτq q. We will will write eτH,σ for this restriction map. 108

Algebraic Representations of Reductive Groups over Local Fields 109 94

Definition 2.3. Let A and B be two R-carapaces on X. The carapace of local homomorphisms Fro A to B will be written HomR,X pA, Bq. Its value on σ is HomR,X pA, Bqpσq “ HomR,Xpσq pA|X pσq, B|X pσqq Its expansions are the maps eτH,σ For want of a direct reference, we include some discussion of the basic properties of Hom. Theorem 2.4. The local homorophism functor, HomR,X pA, Bq, is additive, covariant in B and contravariant in A and left exact in both variables. Moreover (1) HomR,X pRX , Aq » A, functorially in A (2) ΓpX, HomR,X pA, Bqq “ HomR,X pA, Bq (3) There is a canonical isomorphism functorial in A, B, and C, φ : HomR,X pA, HomR,X pB, Cqq Ñ HomR,X pA bR B, Cq (4) There is a functorial isomorphism: HomR,X pRX , HomR,X pA, Bqq » HomR,X pA, Bq Proof. Of the three preliminary statements, only left exactness requires comment. What must be shown is the left exactness of segments of HomR,X pA, Bq as A and B vary over short exact sequences. But HomR,X pA, Bqpσq “ HomR,Xpσq pA|Xpσq , BXpσq q. Restriction to Xpσq is exact and HomR,X pσq is left exact in both of its arguments. The requisite left exactness follows immediately. To establish (1), we must establish the isomorphism on segments. But HomR,X pRX , Aqpσq “ HomR,Xpσq pRX , A|Xpσq q, 109

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but this last expression is equal to: lim Apτq.

ÐÝ τPXpσq

95

However the category, Xpσq has an initial element and so the projective limit is just Apσq. For Statement (2), write: pA|Xpσq , B|Xpσq q ΓpX, HomR,X pA, Bqq “ lim ÐÝ τPX

There is a natural map from ΓpX, HomR,X pA, Bqq to this projective limit. Just send f to the element in the limit whose component at σ is the restriction (the pull-back actually) of f to the sub-category, Xpσq. It is a triviality to verify that this map is an isomorphism. Rather than giving a fully detailed proof of (3), we will give complete definitions of φ and a map ψ which is inverse to it. The necessary verifications, though numerous and quite technical, contain no surprises and so we leave them to the reader. First suppose f P HomR,X pA, HomR,X pB, Cqq. Then for a P Apσq, fσ paq is a family, fσ paq “ t f paqτ uτĚσ where f paqτ P HomR pBpτq, Cpτqq. The following equations express the facts that f is a carapace morphism and that f paq is carapace morphism from B|p σq to C| xpσq : fτ peτA,σ paqqγ “ fσ paqγ γ

(2.1) γ

eC,τ ˝ fτ paqτ “ fτ paqγ ˝ eB,τ

(2.2)

Then we may define φ : HomR,X pA, HomR,X pB, Cqq Ñ HomR,X pA bR B, Cq and ψ opposite to it by the equation: φp f qσ pa b bq “ r fσ paqσ spbq rφpFqσ paqsτ pbq “ fτ preτA,σ paq b bq

a P Apσq, b P Bpσq

(2.3)

a P Apσq, b P Bpτq

(2.4)

These are the two maps, inverse to one another, which establish (3). The last statements is obtained by applying the third to the left hand side and observing that RX bR A » A. 110

Algebraic Representations of Reductive Groups over Local Fields 111

3 Derived Functors In this section we introduce the most elementary derived functors on CarR pXq. These include the derived functors of both the module valued and the carapace valued tensor and homomorphism functors and the relation between the two. First recall that by 1.4, whenever P is a projective R-module and σ is a simplex in X then P Òσ is a projective R-carapace. Consequently a coproduct of carapaces of the form F Òσ , where F is a free R-module, is a projective R-carapace. We will refer to such carapaces as elementary projectives. Furthermore notice that if M is any R-module then σpX, M Òτ q “ M. Hence if Q is an elementary projective, σpX, Qq is a direct sum of free modules and hence free. Lemma 3.1. Let X be any simplicial complex. (1) Every R-carapace on X is a surjective image of an elementary projective. (2) If P is a projective R-carapace on X, then σpX, Pq is R-projective. (3) A tensor product of elementary projectives is an elementary projective. (4) A tensor product of projectives is projective. Proof. Let A be an R-carapace on X. For each σ P X let Fσ be a free R-moduleš and let qσ : Fσ Ñ Apσq be surjective morphism. š Then, just as in 1.5, σPX Fσ Òσ is an elementary projective and σPX Qσ maps in onto A. Thus 1) is established. To prove 2), choose an elementary projective, F, and a surjective map, q : F Ñ P. Since P is projective, it is a direct summand of F and so σpX, Pq is direct summand of σpX, Fq which is free. This establishes 2. The fourth statement follows from the third because, by 1, every projective is a direct summand of an elementary projective. Hence we must prove 3). But this reduces to proving that is σ and τ are two simplices, then R Òσ bR R Òτ is elementary projective. But R Òσ bR R Òτ pγq ‰ 0 111

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if and only if γ Ě σ and γ Ě τ. Thus R Ò bR R Òτ ‰ p0q if and only if σ Y τ “ α is a simplex and then R Òσ bR R Òα . This establishes the result. There are at least four very obvious homological bifunctors on CarR pXq Definition 3.2. Let A and B be R-carapaces on X. Write ExtrR,X for the r’th right derived functor of the left exact module valued bifuncq tor, HomR,X pa, Bq. Write ExtR,X for the carapace valued q’th right derived functor of the carapace valued local homomorhism functor. Write T orqR,X pA, Bq for the q’the carapace valued left derived functor of the carapace valued tensor product, A bR B and write TorR,X q pA, Bq for the q’th left derived functor of the right exact bifunctor σpX, A bR Bq. Lemma 3.3. If P is a projective R-carapace on X, then for any Rcarapace, A, P bR A is σ-acyclic and HomR,X pP, Aq is Γ-acyclic.

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Proof. Let tQ j u, j ď 0 be a projective resolution of A. Then tP bR Q j u is a projective resolution of P bR A. Apply the functor σ to obtain R,X TorR,X j pA, Pq “ H j pA bR Pq. But Tor j pA, Pq “ p0q because P is projective. This establishes the σ-acyclicity of A bR P. For the other acyclicity, let tK j u be a projective resolution of RX . Then HomR,X pK j , HomR,X pP, Aqq “ HomR,x pK j bR P, Aq. But K j b P is a projective j resolution of P. Thus H j pX, HomR,X pP, Aqq “ ExtR,X pP, Aq which is p0q because P is projective. The elementary properties of these four functors are described in the following. Proposition 3.4. Let A and B be R-carapaces on X, Then (1) T orqR,X pA, Bqpσq “ T orqR pApσq, Bpσqq (2) TorR,X q pRX , Aq “ Hq pX, Aq q

(3) ExtR,X pR x , Aq “ H q pX, Aq 112

Algebraic Representations of Reductive Groups over Local Fields 113 p,q

(4) There is a spectral sequence with E2 term: p,q

q

E2 “ H p pX, ExtR,X pA, Bqq and whose abutment is: p`q

ExtX,R pA, Bq (5) There is a spectral sequence with E 2p,q term: E 2p,q “ H p pX, T orqR,X pA, Bqq and with abutment: TorR,X p`q pA, Bq Proof. To prove 1), let P j be a projective resolution of A. Then for each σ, tP j pσqu is a projective resolution of Apσq. Moreover the segment of P j bR B along σ is P j bR bpσq. But the segment along σ is an exact functor on CarR pXq and so the σ-segment of the homology of the complex, P j bR B is the homology of the complex P j pσq bR Bpσq. This is just the result desired. Statements 2 and 3 are both essentially trivial. Just note that q R,X Torq pRX , Aq (respectively ExtR,X pRX , Aq) is a connected sequence of homological functors acyclic on projectives (respectively injectives) and 0 that TorR,X 0 pRX , Aq “ ΣpX, Aq (respectively ExtR,X pRX , Aq “ ΓpX, Aqq. The two statement follow. The local global spectral sequences in 4) and 5) ar just composition of two functor sequences as in [Gr]. Let F A pBq “ A bR B and let G A pBq “ HomR,X pB, Aq. By Lemma 3.3, F A carries projectives to Σ-acyclics and G A carries projectives to Γ-acyclics. The left derived functors of F A are the functors, T orqR,X pA, ´q while the right derived q functors ofG A are the functors, HomR,X pA, ´q. The construction of the spectral sequences is standard. 113

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4 Homological Dimension 98

In this section we will determine the homological dimension of CarR pXq for a finite dimensional simplicial complex, X.ŤWrite X r for the set of simplices in X of dimension r and write Xn for rěn X prq . If M is an Rmodule,write pdpMq for the projective dimension of M and write hdpRq for the homological dimension of R. Definition 4.1. Let A be an R-carapace on X. (1) Write spAq ď n if Apσq “ p0q whenever dimpσq ă n and let spAq “ in f tn : spAq ď nu. Then spAq is called the support dimension of A. (2) We say that A is locally bounded if pdpApσqq ď q for some fixed q ě 0. In that case let ldpAq “ suptpdpApσqq : σ P Xu. When it exists, ldpAq is called the local projective dimension of A. Proposition 4.2. Suppose that A is an R-carapace on X of support dimension at least n and local projective dimension r. Then there is an exact sequence: 0 Ñ A1 Ñ Pr Ñ . . . Ñ P0 Ñ A Ñ 0

(4.3)

so that: (1) ldpA1 q “ 0 (2) Pi is projective (3) spA1 q ě n ` 1 (4) spPi q ě n Proof. For each σ P Xn choose a projective module, Qσ and a surjective σ morphism, φσ : Qσ Ñ Apσq Ñ 0. For each σ, let φš σ : Q Òσ Ñ A be σ σ the morphism š of carapace induced by φ . Let Q0 “ σPXn pQ q Òσ and let d0 “ σPXn φσ . Since spAq ď n, d0 in surjective. Let N0 “ kerpd0 q. Then, clearly spN0 q ď n but ldpN0 q ď r ´ 1. Thus we may repeat the 114

Algebraic Representations of Reductive Groups over Local Fields 115 process the process with A replaced by N0 and continue inductively until we obtain an exact sequence: 0 Ñ Nr´1 Ñ Qr´1 Ñ . . . Ñ Q0 Ñ A Ñ 0. In this sequence, the Qi are projective, the support dimensions of the Qi and of Nr´1 are at least n and ldpNr´1 q are at least n and ldpNr´1 q “ 0. That is Nr´1 pσq is projective for each σ. Now let 99 ž Qr “ pNr´1 pσqq Òσ . σPXn

Clearly, Qr maps onto Nr´1 and the map is an isomorphism on segments over simplices of dimension n. Let dr be the composition of the map onto Nr´1 with the inclusion into Qr´1 and let A1 “ kerpdr q. Clearly, A1 and the Qi answer the requirements of the proposition. Theorem 4.4. Let X be a simplicial complex of dimension d and let A be a locally bounded R-carapace on X of local projective dimension r. Then pdpAq ď d ` r. Proof. Apply Proposition 4.2 with n “ 0. The result is the exact sequence: 0 Ñ A1 Ñ Pr Ñ . . . Ñ P0 Ñ A Ñ 0

Then apply 4.2 to A1 observing that ldpA1 q “ 0 and spA1 q ď 1. The result is a short exact sequence, 0 Ñ A2 Ñ Q1 Ñ A1 Ñ 0 where Q1 is projective, ldpA2 q “ 0 and spA2 q ď 2. We may continue until we reach spAd q ď d. But for any B, if the segments of B are projective and spBq ď dimpXq then B is projective. We may thus assemble these short sequences and the sequence of Pi to obtain a sequence: 0 Ñ Ad Ñ Qd´1 Ñ . . . Ñ Q1 Ñ Pr Ñ . . . Ñ P0 Ñ A Ñ 0 This exact sequence is the projective resolution establishing the result. Corollary 4.5. If dimpXq “ d and if M is an R-module of projective dimension r, then pdpMX q ď r `d. If M is projective then pdpMX q ď d. 115

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Corollary 4.6. If R is of homological dimension r and dimpXq “ d then the homological dimension of CarR pXq is at most d ` r. Neither of these corollaries requires so much as one word of proof.

5 Carapaces and Morphisms of Complexes Recall that a morphism of complexes from X to Y is just a covariant functor from the category of simplices in X to the category of simplices in Y; it is simplicial if it carries vertices to vertices. If S is a subset of the vertex set of X then it admits a simplicial complex structure by taking as its set of simplices the set of simplices in X all fo whose vertices lie in S . We will write S˜ fof this complex. When we speak of a subcategory of X we will always, unless otherwise indicated, mean a full subcategory 100 of the simplex category of X. If U and V are subcategories if X we will write U Ă V to indicate that U is a full subcategory of V. In this case there is always a functorial map from ΣpU, Aq to ΣpV, Aq for any carapace, A. Suppose that f is a morphism of complexes from X to Y and that Z is a simplicial sub-complex of Y. Let f ´1 pZq denote the simplicial subcomplex of X which has as its set of simplices the set tσ P X : f pσq P Zu. Clearly, f ´1 is a functor from the subcomplexes in Y to those in X. Definition 5.1. Let X and Y be simplicial complexes, let A be a Rcarapace on X and let B be one on Y. Let F : X Ñ Y be a morphism of complexes. f pτq

(1) Let p f ˚ pBqqpσq “ Bp f pσqq and let eτf ˚ pBq,σ “ eB, f pσq . Then f ˚ pBq is an R-carapace on X and f ˚ is a covariant functor from CarR pYq to CarR pXq. The carapace f ˚ pBq is called the inverse image of B under f . (2) Let p f˚ pAqqpσq “ Σp f ´1 pσq, ˜ Aq and let eτf

˚ pAq,σ

be the natural

map of segments induces by the inclusion of categories, f ´1 pσq ˜ Ă ´1 f p˜τq when σ Ă τ. Then f˚ is a covariant functor from carapaces 116

Algebraic Representations of Reductive Groups over Local Fields 117 on X to carapaces on Y. The carapaces, f˚ pAq is called the direct image of A by f . Both f ˚ and f˚ are additive. In addition they satisfy the adjointness properties expected. Theorem 5.2. Let X and Y be simplicial complexes, let f be a morphism of complexes from X to Y, let A be an R-carapace on X and let B be one on Y. (1) f ˚ is exact. (2) f˚ is right exact. (3) f˚ is left adjoint to f ˚ . That is, HomR,A pA, f ˚ pBqq » HomR,Y p f˚ pAq, Bq functorially in A and B. Proof. The first statement is a triviality. The second statement in nothing more than the right exactness of co-limits. Thus only the last statement requires attention. To prove 3), we will give morphisms, 101 ψ : HomR,X pA, f ˚ pBqq Ñ HomR,Y p f˚ pAq, Bq and φ inverse to it. Begin with ψ. If α P HomR,X pa, f ˚ pBqq, write α “ tασ uσPX . Then ασ maps Apσq to Bp f pσqq for each σ compatibly with respect to σ. Then σ P f ´1 pρq ˜ if and only if f pσq Ď ρ. Thus the ρ set of maps, eB, f pσq ˝ ασ is direct system of maps giving a morphism from r f˚ pAqspρq “ Σp f ´1 pρq, ˜ Aq to Bpρq. For each ρ call this map βρ . Then since βρ is functorial in ρ, the family tβρ uρPY is a morphism, β, from f˚ pAq to B. Let ψpαq “ β. Now we wish to define φ . If β P HomR,Y p f˚ pAq, Bq then β is a family tBρ uρinY where βρ maps Σp f ´1 pρq, ˜ Aq to Bpρq. For any σ in X, let ρ “ f pσq and let aσ be the natural map from Apσq to Σp f ´1 pρq, ˜ Aq. 117

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Then βρ ˝ aσ is a map from Apσq to Bp f pσqq for each σ. Let ασ “ β f pσq for each σ and let φpβq be the map α “ tασ uσPX . We leave the task of verifying that ψ and φ are maps of the requisite type and that they are inverse to one another to the reader. It is entirely expected the f ˚ has a left adjoint. It is a bit surprising, though not at all subtle, that is also has a right adjoint. Let f : X Ñ Y be a morphism of complexes. For ρ P Y let f : pρq denote the sub-category of X consisting of all σ P X such that f pσq Ě ρ. Definition 5.3. Let X and Y be simplicial complexes, let f : X Ñ Y be a morphism of complexes and let A be an R-carapace on X. Define an R-carapace on Y by the equation: f: pAqpρq “ Γp f : pρq, Aq This is clearly an R-module valued functor on the simplex category of Y and so it is an R-carapace on Y. We will call it the right direct image of A under f . Proposition 5.4. Let f : X Ñ Y be a morphism of complexes, let A be an R-carapace on X and let B be one on Y. Then f: is left exact and right adjoint to f ˚ . That is, HomR,X p f ˚ pBq, Aq » HomR,Y pB, f: pAqq functorially in A and B. 102

Proof. Left exactness follows from the left exactness of Γ and so we only need to establish the adjointness. We give the two morphisms. Let µ : HomR,Y pB, f: pAqq Ñ HomR,X p f ˚ pBq, Aq be one of the two morphisms and let ζ be its inverse. Choose δ in HomR,Y pB, f: pAqq. For each ρ P Y, δ takes each element, b P Bpρq to a compatible family, trδρ pbqsσu f pσqĚρ where rδρ pbqsσ P Apσq. For each σ we must give a map ηpδqσ : Bp f pσqq Ñ Apσq. Let “ ‰ rηpδqσ s pbq “ δ f pσq pbq σ 118

Algebraic Representations of Reductive Groups over Local Fields 119 This defines η. To define ζ, choose b P Bpρq adn suppose that β P HomR,X p f ˚ pBq, Aq. f pσq

If f pσq Ě ρ let aσ “ βσ peB,ρ pbqq. Then let ζpβqρ pbq “ taσ u f pσqĚρ We leave the verifications involved to the reader.

Corollary 5.5. Let X, Y and f be as above. Then: (1) f˚ carries projectives on X to projectives on Y. (2) For any R-carapace A on X, ΣpY, f˚ pAqq “ ΣpX, Aq. (3) f: carries injectives on X to injectives on Y. (4) For any A on X, ΓpY, f: pAqq “ Γpx, Aq Proof. For the first statement, let P be a projective on X and let M Ñ N Ñ 0 be a surjective map in CarR pXq. Consider the map, HomR,Y p f˚ pPq, Mq Ñ HomR,Y p f˚ pPq, Nq. By the adjointness statement in 5.2, 3), this is the same as the map HomR,X pP, f ˚ pMqq Ñ HomR,X pP, f ˚ pNqq. But now f ˚ is exact and P is projective on X and so this map is surjective. This takes care of 1). In general, if M and N are R-modules and there is an isomorphism HomR pM, Qq » HomR pN, Qq functorial in Q, then M » N. Apply this to 2) using the definition of the functor ΣpX, ?q and 3) to obtain: HomR pΣpY, f˚ pAqq, Mq “ HomR,Y p f˚ pAq, MY q “ HomR,X pA, MX q “ HomR pΣpX, Aq, Mq

Statement 2) follows. 103 The proof of 3) is precisely dual to the proof of 1). To establish 4), apply 5.4 and 2.4, 2). Write: ΓpY, f: pAqq “ HomR,Y pRY , f: pAqq “ HomR,X pRX , Aq “ ΓpX, Aq Thus 4) is also proven.

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Corollary 5.5 establishes exactly what is necessary for two composition of functor spectral sequences. Many are possible but we content ourselves with the two most obvious. Proposition 5.6. Let X and Y be simplicial complexes, let A be an Rcarapace on X and let f : X Ñ Y be a morphism of complexes. (1) There is a spectral sequence with E 2p,q term: E 2p,q “ H p pY, Lq f Aq and abutment: Hr pX, Aq p,q

(2) There is a spectral sequence with E2 term: p,q

E2 “ H p pY, Rq f: Aq and abutment: H r pX, Aq These spectral sequences are sufficiently standard that no proof is required. The proofs in [Gr], for example, apply.

6 Certain Special Carapaces This section will be devoted to the study of certain acyclic carapaces. We will need certain conventions. If X is a simplicial, a complement in X is a full subcategory of its simplex category such that the complement of its collection of simplices is a simplicial complex. The reader may verify that C is a complement in X if, whenever σ P C and τ Ě σ then τ P C. Alternatively C is a complement in X if and only if whenever σ P C, then Xpσq Ď C. These two conditions apply to arbitrary subcollections of the simplex set of X and we will use the term complement in this sense. Clearly arbitrary unions and intersetions of complements are complements. 120

Algebraic Representations of Reductive Groups over Local Fields 121 104

Definition 6.1. Let X be a simplicial complex. (1) An R-carapace, B, is called brittle if for every sub-complex of X, Z, the natural map, ΣpZ, Bq Ñ ΣpX, Bq is injective. (2) An R-carapace, F, is called flabby if for every complement in X, C, the natural map ΓpX, Fq Ñ ΓpC, Fq in surjective. Our development follows standard treatments of flabbyness for sheaves. On occasion something more is called for in the brittleness arguments. Flabbyness will be an entirely familiar concept, but brittleness might be a bit strange. We will begin with some descriptive comments. First notice that if dimpXq ą 0 then RX is not brittle. Suppose that σ is positive dimensional simplex in X and that x and y are distinct vertices in it. Let Z “ tx, yu. That is, Z is the disconnected two point complex. Then clearly, ΣpX, RX q “ R ‘ R and, since σ P X and Z Ď σ, the map, ΣpZ, RX q Ñ ΣpX, RX q is not injective since it factors through RX pσq “ R. If σ P X and B is brittle then by definition, Bpσq Ď ΣpX, Bq. But brittleness also forces the relation, BpσqXBpτq “ BpσXτq where the intersection is taken in ΣpX, Bq. To see this just note that, because ΣpZ, Aq is nothing but the inductive limit over Z, there is an exact sequence, ž 0 Ñ Bpσ X τq Ñ Bpσq Bpτq Ñ Σpσ Y τ, Bq Ñ 0

and, by brittleness, an inclusion Σpσ Y τ, Bq Ď ΣpX, Bq. Before proceeding a convention is necessary. If σ is a simplex in X then write σ ˆ for the complex whose vertex set is σ but whose simplex set is the set of all proper subsets of σ. That is σ is not a simplex in σ ˆ which is a simplicial sphere. Then σ ˆ Ă σ. ˜ We will also require the following. Let f : M Ñ N be a morphism of R-modules. Then f is injective if and only if, for each injective Rmodule, J, the induced map HomR pN, Jq Ñ HomR pM, Jq is surjective. Finally suppose that Z is a simplicial sub-complex of X. Let C be the set the simplices of X which are not siplices of Z. For any R-module, M, define R-carapaces MZ˚ and M˚C by the equation: 121

122

William J. Haboush Mz˚ pσq “ M

Mz˚ pσq

if

“ p0q if

σPZ

σRZ

(6.2)

Then M˚C is defined by exactly the same equations, replacing MZ˚ by M˚Z and Z by C. As Z is a complex MZ˚ in naturally a quotient of MX C is naturally a subobject. In fact, the following is exact: 105 while M˚ 0 Ñ M˚C Ñ MX Ñ MZ˚ Ñ 0 In addition, the following hold HomR,X pA, MZ˚ q “ HomR pΣpZ, Aq, Mq

HomR,X pM˚C , Aq “ HomR pM, ΓpC, Aqq

(6.3)

Lemma 6.4. Let X be a simplicial complex, let Z Ď X be a subcomplex of X and let C be a complement in X. Then if A is brittle on X, A|Z is brittle on Z. If A is flabby on X, then A|C is flabby on C. Proof. If A is brittle and Z 1 is subcomplex of Z then the composition, ΣpZ 1 , Aq Ñ ΣpZ, Aq Ñ ΣpX, Aq is the map, ΣpZ 1 , Aq Ñ ΣpX, Aq. If a composition in injective, each map in it injective. This proves the first statement. The proof of the second statement i precisely dual to it and so we leave in to the reader. Theorem 6.5. Let X be a simplicial complex and let 0 Ñ A1 Ñ A Ñ A2 Ñ 0 be exact. (1) If A2 is brittle, then 0 Ñ Σpx, A1 q Ñ ΣpX, Aq Ñ ΣpX, A2 q Ñ 0 is exact. 122

Algebraic Representations of Reductive Groups over Local Fields 123 (2) If A1 is flabby, then 0 Ñ ΓpX, A1 q Ñ ΓpX, Aq Ñ ΓpX, A2 q Ñ 0 is exact. 106

Proof. To prove (1) we need only show that ΣpX, A1 q Ñ ΣpX, Aq is injective. By the observation above, it would suffice to show that HomR pΣpX, Aq, Jq Ñ HomR pΣpX, A1 q, Jq is surjective for an injective, J. But HomR pΣpX, Aq, Jq “ HomR,X pA, JX q and the same for A1 . Thus, to establish (1), it suffices to prove that every carapace morphism, f : A1 Ñ JX extends to a morphism. f˜ : A Ñ JX . Let j : A1 Ñ A be the injection and let π : A Ñ A2 be the surjection. Let f : A1 Ñ JX be a morphism of carapaces. Let F be the family of paris, pZ, fZ q where Z is a subcomplex and fZ : A|Z Ñ JZ is a morphism such that fZ ˝ J “ f |Z . Order these by inclusion on Z and extension on fZ . This orders F inductively and so Zorn’s Lemma yields a maximal element, pW, fW q. If W ‰ X there is some σ P X such that σ R W. If σ X W “ H we may trivially extend fW to W Y ttu where t is any vertex in σ. This contradicts maximality. Thus we may assume that σ X W ‰ H. Let σ ˜ X W “ Y. Consider fσ : A1 pσq Ñ J. By the injectivity of J, we may choose fσ1 : Apσq Ñ J such that fσ1 ˝ jσ “ fσ . If γ Ď σ let fγ1 “ fγ1 ˝ eσA,γ . Since j is morphism, the following commutes: A1 pσq eσ1

jσ

O

eσA,γ A

A ,γ

A1 pγq

/ Apσq O

jγ

/ Apγq

Hence fγ1 ˝ jγ “ fσ1 ˝ eσA,γ ˝ jγ “ fσ1 ˝ eσA1 ,γ “ fγ . That is, f 1 ˝ j “ f on σ. But on σ ˜ X W “ Y, fW ˝ j “ f . Thus on Y, p fW ´ f 1 q ˝ j “ 0. It follows that fW ´ f 1 induces a map form A2 |Y to JY . But ΣpY, A2 q Ñ pX, A2 q is brittle. Hence HomR,X pA2 , J x q Ñ HomR,Y pA2 |Y, JY q is surjective. Thus, there is and f2 P HomR,X pA2 , JX q such that π˝ f2 |Y “ p fW ´ f 1 q|Y . Consequently, p f 1 ` π ˝ f2 q|σXW “ fW |σXW . Hence fW can be extended ˜ ˜ 123

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William J. Haboush

to W Y σ ˜ contradicting the maximality of pW, fW q. That is W “ X and so 1) is established. To prove 2) we must prove that ΓpX, Aq Ñ ΓpX, A2 q is surjective. An element a P Γpz, Aq is a function on Z such that apσq P Apσq and eτA,σ papσqq “ apτq. Suppose a2 P ΓpX, A2 q is given. Order the pairs pC, aC q, where C is a complement and aC P ΓpC, Aq, πpac q “ a2 |C , by inclusion and extension. This being an inductive order, there is a maximal element, pU, aU q. If U ‰ X, there is simplex, τ not in U. Choose a˜ r P Apτq such that πτ p˜ar q “ a2 pτq. Define a˜ 1 in Xpτq by ˜ r q. If Xpτq X U “ H then a˜ 1 extends aU contradicting a˜ 1 pσq “ eσA,τ ppaq maximality of pU, aU q, and so we may assume that Xpτq X U ‰ H. This intersection is a complement. Consider the difference a˜ 1 ´ aU on ˜ 1´ this intersection. Now πp˜a1 ´ aU q “ 0 on Xpτq X U whence pa 1 1 aU q|Xpτq X U P ΓpXpτq X U, A q Since A is flabby there is an element a1 P ΓpX, A1 q such that a1 |Xpτq X U “ pa1 ´ aU q|Xpτq X U. Clearly 107 a1 ´ pa1 |Xpτqq extends aU contradicting maximality. Thus U “ X and we have established 2) Corollary 6.6. Let 0 Ñ A1 Ñ A Ñ A2 Ñ 0 be an exact sequence of R-carapaces on X. (1) If A and A2 are brittle, then A1 is also. (2) If A and A1 are flabby, then A2 is also. Proof. We prove 1). Suppose Z is a subcomplex of X. Then, by 6.4, A|Z and A2 |Z are both brittle and hence, 0 Ñ ΣpZ, A1 q Ñ ΣpZ, Aq is exact. Thus the following diagram, which has exact rows and columns, commutes: 0 0

/ ΣpZ, A1 q

0

/ ΣpX, A1 q

124

/ ΣpZ, Aq / ΣpX, Aq

Algebraic Representations of Reductive Groups over Local Fields 125 It is immediate the ΣpZ, A1 q Ñ ΣpX, A1 q is monic. As for Statement 2), noting that Z must be replaces by a complement, the proof is both well known and strictly dual to the proof of 1). Proposition 6.7. Let X and Y be simplicial complexes let f : X Ñ Y be a morphism of complexes and let A be an R-carapace on X. Then (1) If A is brittle, then f˚ pAq is brittle. (2) If A is flabby then f: pAq is flabby. Proof. To prove 1), let U Ď Y be a subcomplex. Then, f ´1 pUq is a subcomplex of X and so, if A is brittle, then Σp f ´1 pUq, Aq Ñ ΣpX, Aq is injective. But Σp f ´1 pUq, Aq “ ΣpU, f˚ Aq and ΣX, A “ ΣpY, f˚ Aq by definition. That proves the first statement. The proof of 2) is completely parallel except that it uses 5.5, 4 in place of the corresponding properties of f˚ Proposition 6.8. Let X be a simlicial complex. (1) Projective carapaces are brittle; injective carapaces are flabby. (2) a coproduct of brittle carapaces is brittle; a product of flabby carapaces is flabby. (3) For any simplex, σ P X and any R-module, M, M Òσ is brittle and M Óσ is flabby. Proof. Let Z be any subcomplex of X. let C be its complement and let M any R-module. Then 0 Ñ M˚C Ñ MX Ñ MZ˚ Ñ 0 is exact. Thus, if P is projective, HomR,X pP, MX q Ñ HomR,X pP, MZ˚ q is surjective, But this is the map, HomR pΣpP, Xq, Mq Ñ HomR pΣpP, Zq, Mq. But this map will be surjective for every M if and only if the map ΣpZ, Pq Ñ ΣpX, Pq is injective (in fact, it must be split). If I in injective, we need only consider the case, M “ R. Then HomR,X pRX , Iq Ñ HomR,X pRC˚ , Iq is surjective. This is the sequence, ΓpX, Iq Ñ ΓpC, Iq and hence (2) is established. 125

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To prove 2), let tAi uiPI be a family of R-carapaces on X. Since inductive limits of arbitrary co-products are co-products and projective limits of products are products, we may write: ˜

Σ Z, ˜

Γ C,

ž

Ai

iPI

ź iPI

Ai

¸

¸

“ “

ž iPI

ź iPI

ΣpZ, Ai q (6.9) ΓpC, Ai q

Since a co-product of monomorphisms is monic and a product of surjections is surjective, 3) follows at once. Statement 3) is quite clear. Proposition 6.10. If A a brittle R-carapace on X, then Hi pX, Aq “ 0 for all i ą 0. If F is flabby, then H i pX, Fq “ 0 for all i ą 0. Proof. First choose a projective, P and a surjective map so that there is an exact sequence: 0 Ñ A0 Ñ P Ñ A Ñ 0 The acyclicity of P the Theorem 6.5 together imply that for any brittle A, H1 pX, Aq “ 0. Then choose a projective resolution of A. Break 109 this into a series of short exact sequences, use Corollary 6.6 and apply induction. The same technique, applied dually, gives the second statement. We conclude with local criteria for which there no immediate applications but which are somewhat interesting. Proposition 6.11. Let A a be an R-carapace on X. (1) If for each simplex σ P X the map, Σpσ, ˆ Aq Ñ Σpσ, ˆ Aq, is injective then A is brittle. (2) If for each simplex σ P X the restriction A|Xpσq is flabby, then A is flabby. 126

Algebraic Representations of Reductive Groups over Local Fields 127 Proof. Proofs of these statements use Zorn’s lemma as its was used in Theorem 6.5 First we prove 1). Let Z be an arbitary subcomplex of X. We must show that ΣpZ, Aq Ñ ΣX, A is injective. As in the proof of theorem 6.5, this comes to proving that for any injective module, J, any morphism, f : A|Z Ñ JZ , admits and extension, f : A Ñ JX . Applying Zorn one finds a maximal subcomplex on which f admits and extension and so, replacing Z by this maximal subcomplex, we may assume that f does not extend to any subcomplex contating Z. If the vertex x is not in Z then f clearly extends to the disconnected union and so we may assume that very vertex is in Z. Choose a simplex, σ of minimal dimension among the simplices not in Z. Then σ ˆ Ď Z. Making use of the condition in (1), we obtain a diagram: 0

/ Σpσ, ˆ Aq

/ Σpσ, ˜ Aq

ΣpZ, Aq fZ

J Hence there is a map, fi : Σpσ, ˜ Aq Ñ J extending fZ and so one may extend f to Z Y σ contradicting maximality. It follows that it must be that Z “ X. The proof of 2), by duality, in entirely straightforward and so we omit it.

7 Canonical Resolutions In this section we give canonical chain and co-chain complexes which can be used to compute the exoskeletal homology and cohomology groups. They arise from canonical resolutions and they are sufficiently 110 canonical that they will be seen to be equivariant when there is a group action involved.

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Let A be an R-carapace on X. Then by 1.4, 1) and 2), the identity map on Apσq induces a map, πσ : pApσqq Òσ Ñ A and a map jσ : A Ñ pApσqq Óσ . Definition 7.1. Let X be a simplicial complex and let A be an R-carapace on X. (1) Let T0 pAq “ (2) Let S0 pAq “

ž

pApσqq Òσ

and let πA “

ź

pApσqq Óσ

and let

σPX

σPX

jA “

ž

πσ

ź

jσ .

σPX

σPX

(3) Let K0 pAq “ KerpπA q. (4) Let C0 pAq “ Cokerp jA q. This definition has certain immediate consequences. Lemma 7.2. Let X be a simplicial complex and let A be an R- carapace on X. (1) The four functors, T0 , K0 , S0 , and C0 are exact additive functors. (2) Both πA and jA are natural transformations in the argument A. Further πA is always surjective and jA is always monic. (3) For all A, T0 pAq is brittle and S0 pAq is flabby. (4) If A is brittle, then K0 pAq is brittle; if A is flabby, C0 pAq is flabby.

Proof. That T0 and S0 are exact and additive is a trivial observation. Since T0 pAq is a coproduct of carapaces of the form M Òσ , proposition 6.8, 2) and 3) guarantee that it is brittle. The flabbyness of S0 pAq follows similarly from the fact that it is a product of carapaces of the form M Óσ . That K0 and C0 are exact is littel more thant the snake lemma. Statement 2) is a triviality and so only 4) remains to be proven. This follows from 3) and Corollary 6.6. 128

Algebraic Representations of Reductive Groups over Local Fields 129 Definition 7.1 and lemma 7.2 are just what is necessary to construct standard resolutions. Definition 7.3. Let A be an R-carapace on X. Let Kn pAq “ K0 pKn´1 pAqq 111 and let Cn pAq “ C0 pCn´1 pAqq. That is Kn is the pn ` 1q1 st iterate of K0 and the same, mutatis mutandis, is ture for Cn . Let Tn`1 pAq “ T0 pKn pAqq and let Sn`1 pAq “ S 0 pCn pAqq for n ď 0. Define maps, δn : Tn`1 pAq Ñ Tn pAq and δn : Sn`1 pAq as follows. The map, δn is the composition of the natural surjection, Tn`1 pAq Ñ\ pAq, with the inclusion, Kn pAq ãÑ Tn . Similarly δn is the composition of the surjection, Sn pAq Ñ C n pAq, and the inclusion, Cn pAq ãÑ Sn`1 pAq. Then tTn pAq, δn u is called the canonical brittle resolution and tSn pA, δn qu is called the canonical flabby resolution of A. Some remarks are in order. First of all, since each of the functors, Tn and Sn , are compositions of exact functors, they are themselves exact functors. Further, by Lemma 7.2, for any A, each of the carapaces Tn pAq is brittle while the Sn pAq are flabby. Thus, letting Cn pX, Aq “ ΣpX, Tn pAqq and C n pX, Aq “ ΓpX, Sn pAqq, whenever 0 Ñ A1 Ñ A Ñ A2 Ñ 0 is exact, 0 Ñ Cn pX, A1 q Ñ Cn pX, Aq Ñ Cn pX, A2 q Ñ 0 and 0 Ñ C n pX, A1 q Ñ C n pX, Aq Ñ C n pX, A2 q Ñ 0 are exact. Abusing language, use δn and δn for the maps of segments and sections respectively as well as maps of carapaces, the homology groups of the complexes, tCn pS , Aq, δn u and tC n pX, Aq, δn u are connected sequences of homological functors. Definition 7.4. Let A be an R-carapace on X. The complex, tCn pX, Aq, δn u will be called the complex of Alexander chains on X with coefficients in A; C n pX, Aq, δn u will be called the Alexander co-chains. The homology of the complex of Alexander chains will be called the Alexander homology and it will be written, Hna pX, Aq. The homology of the Alexander co-chain complex will be called the Alexander cohomology and it will be written Han pX, Aq. 129

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Proposition 7.5. The Alexander homology and cohomology of the simplicial complex, X, with coefficients in A are isomorphic, respectively, to the exoskeletal homology and cohomology of X with coefficients in A, functorially in A. Proof. By Proposition 6.10, the exoskeletal homology groups vanish on brittle carapaces while the cohomology groups vanish on flabby carapaces. Hence the Alexander groups are the homology groups of the segments (respectively sections) over an acyclic resolution. The proposition follows. The following is an interesting footnote.

112

Proposition 7.6. If A is projective, the canonical brittle resolution of A consists of projective carapaces. If A in injective, each term in the canonical flabby resolution in injective. Proof. It suffices to prove that if A is projective then T0 pAq and K0 pAq are projective and the corresponding statement for an injective A and S0 and C0 . Suppose that P is projective and that I in injective. Then by Proposition 1.6, Ppσq is projective and Ipσq is injective for each σ P X. But then, by 1.4, pPpσqq Òσ is projective and pIpσqq Óσ in injective. By the definition of T0 and S 0 and because coproducts of projective are projective and products of injectives are injective, T0 pPq is projective and S0 pIq is injective. But then 0 Ñ K0 pPq Ñ T0 pPqPπP Ñ P Ñ 0 and j

I 0ÑIÝ Ñ S0 pIq Ñ C0 pIq Ñ 0

are exact. The last two terms of the first sequence are projective while the first two terms of the second sequence are injective. Hence K0 pPq in projective and C0 pIq is injective. An iterative application of these facts establishes the result. 130

Algebraic Representations of Reductive Groups over Local Fields 131

8 G-carapaces and their Homology In this section we consider a simplicial complex, X, with a G-action for some group, G. Then there is a corresponding notion of G-carapace and several ways of constructing G-representations on the homology and cohomology of a G-carapace. One of our main purpose in this section is to show that all of these representations are the same. The method is standard “relative homological algebra”. If X is a simplicial complex and G is group, a simplicial action of G on X is an action of G in the vertex set of X which carries simplices to simplicies. If σ is a simplex in X, write Gσ for the setwise stabilizer of σ and Gˆ σ for the pointwise stabilizer of σ. We will usually write tg for the translation map, tg pxq “ gx. Then, if A is an R-carapace on X, the iverse image of A under tg is the carapace, rtg pAqspσq “ Apgσq. When space does not permit otherwise, write g˚ A for tg˚ pAq. Recall gτ that the expansions on tg˚ pAq are the maps, eτg˚ A,σ “ eA,gσ . Notice that tg˚ pth˚ Aq “ g˚hg pAq. Definition 8.1. A G-carapace on X is an R-carapace, A together with 113 a family of isomorphisms, Φ “ tΦg ugPG , Φg :Ñ tg˚ A, such that for any pair, g, h P G, the diagram: A

Φg

/ t˚ A g tg˚ Φh

Φhg

(8.2)

tg˚ pth˚ Aq

˚ A thg

commute. That is, tg˚ pΦh q ˝ Φg “ Φhg . If pA, Φq and pB, Ψq are two G-carapaces a G´morphism f : A Ñ B is just a morphism such that Ψg ˝ f “ tg˚ p f q ˝ Φg . Clearly G-carapaces are an Abelian category. We now apologize for a digression ś which some would ś perfer to conceal in an “obviously”. Write A for the functor σPX Apσq. An ś element, a, in A is a function such that apσq P Apσq. If α is an auto131

132

William J. Haboush

ś ˚ ś α : α A Ñ morphism of X, there is an isomorphism, I A “ α ‰ š A. It is de´1 fined by IA paq pσq “ apα pσqq. šNow the coproduct, σPX Apσq, the module of relations, N,ś such that σPX Apσq{N “ ΣpX, Aq and ΓpX, Aq are all submodules of A preserved by IAα and hence Iαα induces two other maps, both of which we will denote IAα , from ΣpX, α˚ Aq to ΣX, A and from ΓpX, α˚ Aq to ΓpX, Aq. These maps areś functorial in the same ways and so we will describe their properties for A and consider them established for all three functors. Let A and B be two R-carapaces and let α and β be automorphisms of X. Let φ : A Ñ B be a morphism. The following equations, whose proof we leave to the reader, are the properties of interest. śWe emphasize that we shall use these equation for Σ and Γ rather than . β α˝β I α ˝ I ˚ “ IA ´źA ¯ α A ź (8.3) φ ˝ IAα “ IBα ˝ α˚ φ

114

In general, if f : A Ñ B is a morphism, write f Σ and f Γ for the induced morphisms on the segments and the sections respectively. Let pA, Φq be a G-carapace on X. Then there are natural representations of G on ΣpX, Aq and ΓpX, Aq respectively denoted ΦΣ and ΦΓ defined by the equations: ` Σ˘ g Φ g “ IA ˝ pΦg qΣ ` Γ˘ (8.4) g Φ g “ IA ˝ pΦg qΓ

To see that these are representation, we just apply 8.3 Then, pΦΣ qgh “ g gh g g˚ A IA ˝pΦgh qΣ “ IA ˝Ih ˝pph˚ Φpgq qΣ q˝˝ΦΣh “ IA ˝pIg˚ A ˝ph˚ Φg qΣ q˝ΦΣh “ g Σ Σ Σ h Σ IA ˝ pΦg ˝ IA q ˝ Φh “ pΦ qg ˝ pΦ qh The computation for ΦΓ is virtually identical. It is also clear that this argument gives canonically determined representations of G on the left derived functors of ΣpX, ?q and right derived functors of ΓpX, ?q. We give the argument for ΣpX, ?q. Let α be an automorphism of X. Then α˚ is an automorphism of CarR pXq and so it carries projectives to projectives and injectives. It is moreover an exact functor. Let . . . Ñ Pr Ñ Pr´1 Ñ P0 Ñ A Ñ 0 be a projective resolution of A. Then . . . Ñ α˚ Pr Ñ α˚ Pr´1 Ñ . . . Ñ α˚ P0 Ñ α˚ A Ñ 0 132

Algebraic Representations of Reductive Groups over Local Fields 133 is projective resolution of A and, applying ΣpX, ´q, deleting α˚ A and taking homology yields the left derived functors of ΣpX, α˚ p´qq. Making use of the functoriality expressed by the second equation of 8.3), we obtain a commutative diagram. ¨¨¨

/ ΣpX, α˚ Pr q

/ ¨¨¨

/ ΣpX, Pr q

/ ΣpX, α˚ P0 q IPα

I αpr´1

I αpr

¨¨¨

/ ΣpX, α˚ Pr´1 q

0

/ ΣpX, Pr´1 q

/ ¨¨¨

/ ΣpX, P0 q

Passing to the homology of these complexes, we obtain unique, canonically defined morphisms, Lr IAα : Hr pX, α˚ Aq Ñ Hr pX, Aq. Clearly, the dual construction will yield canonical morphisms, Rq IAα : H q pX, α˚ Aq Ñ H q pX, Aq. Suppose that G acts on X and that A is a G-carapace with G-structure, Φ. It is now clear that there is a canonical representation of G on the exoskeletal homology and cohomology groups of X in A. Write Lr Φg : Hr pX, Aq Ñ Hr pX, g˚ Aq and Rq Φg : H q pX, Aq Ñ H q pX, g˚ Aq for the map induced by Φg on the homology and the cohomology req g g g g spectively. Let Φg “ Rq IA ˝ Rq Φg and Φr “ Lr Ia ˝ Lr Φg . Then Φr and q Φg are easily seen to give the unique representations on the homology and cohomology groups making them into homological functors with values in the category of G-modules. What remains in the question of natural G-structure on carapace valued functors applied to G-carapaces. Let A be a carapace with Gstructure, Φ, and let B be one with G-structure, ψ. First consider the tensor product, A bR B. Inverse image preserves tensor product. That 115 is, tg pA bR Bq » tg˚ pAq bR tg˚ pBq. Consequently, the family of isomorphisms, tΦg bR ψg : g P Gu, is a G-structure on A bR B. observe that the commutativity ex-pressed by diagram 8.2) can also be described by the equation: Φh,gσ ˝ Φg,σ “ Φhg,σ (8.5) Consider the carapace of local homomorphisms. By definition HomX,R pA, Bqpσq “ HomXpσq,R pA|Xpσq , B|X pσqq. We define a map, Θg,σ : HomXpσq,R pA|Xpσq , B|Xpσq q Ñ HomXpgσq,R pA|Xppgqσq , B|Xpgσq q. 133

134

William J. Haboush

The translation, tg maps Xpσq to Xpgσq. Hence Φg maps A|Xpσq to tg˚ pA|Xpgσq q and similarly for B. Consequently, for any f P HomXpσq,R pA|Xpσq , B|Xpσq q, ψg,,σ ˝ f ˝ pΦg,σ q´1 maps tg˚ pA|Xpgσq q to tg˚ pB|Xpgσq q. Hence, tg˚´1 pψg,σ ˝ f ˝ pΦg,σ q´1 q P HomXpgσq,R pA|Xpgσq , B|Xpgσq q. But, this last group is just HomX,R pA, Bqpgσq. Hence, define the G-structure on HomX,R by the equation: Θg,σ p f q “ tg˚´1 pψg |X pσq ˝ f ˝ pΦg |X pσqq´1 q

(8.6)

This equation is to be understood in the following sense. The map Φg |Xpsigmaq maps the restriction of A to Xpσq to the corresponding restriction of tg˚ A while ψg |Xpσq does the same for B. Hence the composition in parentheses takes tg˚ pAq| xpσq to tg˚ pBq| xpσq . Thus the inverse image of this map under tg´1 yields an element of HomX,R pgσq “ HomXpgσq,R pA|Xpg σq , B|Xpgσq q which is what i needed. We check that Θ is a G-structure by establishing 8.5 for it by direct computation. The computation is: ´1 ˚ q Θh,g,σ p f q “ tphgq ´1 pψhg ˝ f ˝ pΦhg q

´1 ˚ “ th˚´1 ptg˚´1 ptg˚´1 tg˚ pψg q ˝ f ˝ Φ´1 g ˝ tg pΦh qqq

´1 “ th˚´1 pψh ˝ tg´1 f ˝ Φ´1 g q ˝ Φh q

“ Θh,gσ pΘg,σ p f qq

Thus Θ is a natural G-structure on HomX,R pA, Bq. But ΓpX, HomX,R pA, Bqq “ HomX,R pA, Bq. Hence 8.4 determines a representation of G on HomX,R pA, Bq. This is what we will call the natural representation of G on HomX,R pA, Bq. The explicit description of this action is: g ¨ f “ tg˚´1 pψg ˝ f ˝ pΦg q´1 q

gPG 134

f P Hom x,R pA, Bq

(8.7)

Algebraic Representations of Reductive Groups over Local Fields 135 116

Establish this as follows. If T is and G-carapace with G-structure Θ, then if τ P ΓpX, T q the action of G on ΓpX, T q defined by 8.4 is described by the equation: pg.τqσ “ Θg,g´1 σ pτg´1 σ q (8.8) When T “ HomX,R pA, Bq, this becomes pg ¨ f qσ “ Θg,g´1 σ p fg´1 σ q. Under the identification of HomX,R pA, Bq with ΓpX, HomX,R pA, Bqq, the σ component of the map, f , is f |Xpσq . Using this, apply 8.6 to compute the right hand side of 8.8 for T “ HomX,R pA, Bq. We obtain: pg ¨ f qσ “ tg˚´1 pψg |Xpg´1 σq ˝ p f |Xpg´1 σq q ˝ pΦg |Xpg´1 σq q´1 q. By the definition of inverse image, this is ptg˚´1 pψg´1 ˝ f ˝ pΦg q´1 qq|Xpσq and this is just the right hand side of 8.7 restricted to Xpσq. This proves the truth of 8.7. Proposition 8.9. Let A, B and C be G-carapaces with G-structures, Φ, ψ and Υ respectively. Then, the natural adjointness isomorphism φ : HomX,R pA, HomX,R pB, Cqq Ñ HomX,R pA bR B, Cq is a G-morphism. Proof. Fix σ P X, A P Apσq, b P Bpσq and f P HomX,R pHom x,R pB, Cqq and recall the definition of φ. It is φp f qσ pa b bq “ r fσ paqsσ pbq and in interpreting this formula one must remember that fσ paq P HomX,pσq,R pB|Xpσq , C|Xpσqq. We will show that φpg¨ f q “ g¨φp f q and we will prove this by evaluating both sides of this equation on a a b b P Apσq b Bpσq. Starting with the left hand side: rφpGg ¨ f qσ spa b bq

˚ ´1 “ rptGg qqσ paqsσ pbq by 8.7 ´1 pΘGg ˝ f ˝ pΦGg q

“ rpΘGg,Gg´1 σ ˝ fGg´1 σ ˝ pΦGg,Gg´1 σ q´1 qpaqsσ pbq

“ ΥGg,Gg´1 σ pr fg´1 σ pΦ´1

Gg,Gg´1 σ

paqqGg´1 σ spψGg,Gg´1 σ pbqqq

“ ΥGg,Gg´1 σ rφp f qpΦGg,Gg´1 σ paq b φGg,Gg´1 σ pbqqs 135

136

William J. Haboush by the def. of ψ “ rGg ¨φp f qspa b bq by 8.7q

That is φpGg ¨ f qσ pa b bq “ rGg ¨φp f qsσ pa b bq as asserted. This proves the result. We conclude this section by showing that the canonical brittle and flabby resolutions of a G-carapace, A are naturally equivariant. A cosideration of the definition of these resolutions shows that if suffices to show that there are canonical G-structure on T0 pAq and S0 pAq so that the maps, T0 pA Ñ Aq and A Ñ S0 pAq are G-equivariant. 117

Lemma 8.10. Let α : X Ñ X be an automorphism. Let M be an R-module, let A be an R-carapace and let Φ : A Ñ α˚ A be an isomorphism. (1) α˚ pM Òσ q “ M Òα´1 σ . ´1 σ

(2) α˚ pM Óσ q “ M Óα

.

(3) There is a natural equality T0 pα˚ Aq “ α˚ f pT0 pAqq. (4) There is a natural equality S0 pα˚ Aq “ α˚ S0 pAq. Proof. The first two statements are trivially true. As for the third and fourth statements, the proofs are nearly identical and so we prove only the first. Write: ž T0 pα˚ Aq “ α˚ pAqpσq Òσ σPX

ž

“

σPX

pApασqq Òσ

“

σPX

pApαq Òα´1 σ

“

ž

ž

σPX ˚

α˚ pApσq Òσ q

“ α T0 pAq 136

Algebraic Representations of Reductive Groups over Local Fields 137 Thus α˚ pT0 pAq “ T0 pα˚ Aqq. That is, the two apparently different construction applies to A, α˚ T0 pAq and T0 pα˚ pAqq result in identically the same object. Proposition 8.11. Let A be a G-carapace on X with G-structure Φ. For each Gg P G, let ΦTGg “ T0 pΦGg q and let ΦSGg “ S0 pΦGg q. Then ΦT and ΦS are G-structures. Furthermore the surjection, πA : T0 pAq Ñ A and the injection, jA : A Ñ S0 pAq are G-equivariant. Proof. First, we show that ΦT and ΦS are G-structures. For ΦT the calculation is: ΦTGg “ T0 pth˚ pΦGg q ˝ Φh q

“ T0 pth˚ pΦGg qq ˝ T0 pΦh q

“ th˚ pT0 pΦGg qq ˝ T0 pΦh q

To prove that πA and CMjmathA are equivariant just note that π is a natural transformation from T0 to the indentity functor while j is one form the identity functor to S0 . Then note that ΦTGg and ΦSGg are just the values of T0 and S0 on the morphism, ΦGg . Corollary 8.12. Let A be a G-carapace on X. Then there are canonical G-structures on the canonical brittle and flabby resolutions of A so that the natural morphisms are G-equivariant. Proof. This is nothing more than an interative application of 8.11). The 118 details are left to the reader.

9 Induced and Co-Induced Carapaces Let X be a simplicial complex. Recall that Xprq denotes the set of simplices of dimension r. We will use Xn to denote Ť the collection of simplices of dimension at least n. That is, Xn “ rěn Xprq. Let G be a group acting simplicially on X. Recall that for any simplex, σ P X, Gσ “ tGg : Gg P G, Gg σ “ σu and Gˆ σ “ tGg : Gg P 137

138

William J. Haboush

Gσ Gg x “ x @ P σu. We will call the action of G on X separated if whenever τ Ď σ and Gg τ Ď σ for some Gg P G, then Gg τ “ τ. If G acts on X, let Yp0q “ xp0q{G be the orbit space and let π : Xp0q Ñ Yp0q be the quotient map. Construct a simplicial complex, Y, with vertex set, Yp0q, by taking as simplices in Y all finite subsets, τ Ď Y such that τ “ πpσq for some simplex σ in X. If the action of G on X is separated, then for each simplex σ in X, Gσ Ď Gˆ σ and the dimension of πpσq is equal to the dimension of σ. If G is acting on X so that the action is separated and if Y is the quotient with quotient map π : X Ñ Y, then a section to π is a simplicial map, s : Y Ñ X, such that π ˝ s “ idY . A separated action admitting a section, s : Y Ñ X, will be called an excellent action. If the action of G on X is excellent with section, s : Y Ñ X, we will identify Y with its image, spYq, in X and we shall refer to π as the retraction onto Y. We will describe this situation by saying that pX, Gq is an excellent pair with retraction π :Ñ Y. Notice that the action of G on X is separated if and only if whenever Xpσq X XpGg σq ‰ H then Gg σ “ σ. If C is a category and X is a simplicial complex, then a C-valued sheaf on X is just a contravariant functor from X to C. If σ Ď τ and S is a σ : Spτq Ñ Spσq for the corresponding map and call sheaf on X write rS,τ it the restriction. If G operates simplicially on X, then the assignment, ˆ Gpσq “ Gˆ σ is a sheaf of groups on X. If the action is separated, then Gσ “ Gˆ sigma , and so this also is a sheaf of groups. In any case we will refer to Gˆ as teh stabilizer of the action. If f : X Ñ Y is a morphism of complexes and S is a sheaf on Y then f ˚ S, defined by the equation f ˚ Spσq “ Sp f pσqq with the corresponding restrictions is called the inverse image of S. When f is the inclusion of a subcomplex, we call f ˚ S the restriction of S to X and we may on occasion write it, SX . Suppose now that G is a group and that H is a subgroup. We wish to fix notation for induced and co-induced modules. Write RrGs for the 119 group algebra of G over R and write RrGsℓ for the free rank one RrGsmodule isomorphic to RrGs as an R-module but with RrGs structure defined by the equation, Gg ¨x “ x Gg´1 for x P RrGs and Gg P G. We simply write Gg x for the product in RrGs. Let M be an H-module. 138

Algebraic Representations of Reductive Groups over Local Fields 139 Then the G-module induced by M is RrGs bRrHs M; the G-module co-induced by M is HomRrHs pRrGsℓ , Mq. The G-structure on the induced module is just that obtained from left multiplication on RrGs. The structure on the co-induced module is just the structure described by pGg f qpxq “ f pGg´1 xq. Write IG{H M for the induced module and CG{H M for the co-induced module. Choose a complete set of coset representatives Q Ă G for the space š of left cosets, G{H. Then IG{H M “ GgPQ Gg bM. Write Gg M for Gg bM. The R-module, Gg M depends only on the coset Gg H and not on the particular representative, Gg. Suppose now that H “ Gσ for some simplex, σ. If γ P Gσ write γ M to denote xM for any x such that xσ “ γ. Then M γ depends only on γ for xσ “ yσ “ γ if and only ifš x P yGσ “ yH. š If Gg γ “ τ then Gg M γ “ Mτ. ś Dually, RrGsℓ “ xPQ RrHs ¨ x “ xPQ xRrHs. Hence C xPQ pMq “ xPQ HomRrHs pxRrHs, Mq. If γ P Gσ let Mγ “ HomRrHs pxRrHs, Mq for any x such that xσ “ γ. This is well defined. Moreover, if yγ “ λ then left translation by y carries Mγ to Mλ . Definition 9.1. Suppose G acts on X, that σ is a simplex in X and that M is a Gσ representation over R. Let: ž T σ pMq “ M τ Òτ rPGσ

σ

S pMq “

ź

rPGσ

M τ Óτ

(9.2)

Then T σ pMq is called the carapace induced by M and S σ pMq is called the carapace coinduced by M. Proposition 9.3. Let G act on X and let M be a representation of Gσ over R. Then (1) T σ M and S σ M both admit canonical G-structures. (2) Let A be any G-carapace on X. Then HomX,R pT σ M, AqG “ HomGσ pM, Apσqq 139

140

William J. Haboush and Hom x,R pA, S σ MqG “ HomGσ pApσq, Mq

(3) T σ pMq is brittle; S σ pMq is flabby. 120

Proof. First observe that 3) is just a consequence of Definition 9.1 and Proposition 6.8. We pass to the construction of G-structures on T σ pMq and S σ pMq. First we evaluate these carapaces on a typical simplex, τ. Recall that τ˜ denotes the full simplicial complex underlying τ. Then: ž T σ pMqpτq “ Mγ γP˜τXGσ

S σ pMqpτq “

ź

(9.4)

Mγ

γPXpτqXGσ

If Gg P G, then Gg carries distinct simplicies in τ˜ X Gσ to disĄτ X Gσ. Hence left multiplication by Gg cartinct simplices in Gg ries separate summands in T σ pMqpτq to the corresponding summands in T σ pMqpGg τq. Let ΦGg,τ be the sum of left multiplication by Gg on the separate components of T σ pMqpτq. Similarly define a map, ψGg,τ , from S σ pMqpτq to S σ pMqpGg τq by taking it to be left translation by Gg on each of the factors. Then, using Equation 8.5, one verifies that Φ and ψ are G-structures. Only 2) remains to be proved. We compute directly. ž HomX,R pT σ pMq, Aq “ HomX,R p M τ , Aq “

“

ź

τPGσ

τPGσ

ź

τPGσ

HomX,R pM τ Òτ , Aq

HomR pM τ , Apτqq

by4, (1)of1

Let Υ “ tΥ śGg uGgPG be theτ G-structure on A and let t fτ uτPGσ be an element of τPGσ HomR pM , Apτqq. The element fτ may be thought of 140

Algebraic Representations of Reductive Groups over Local Fields 141 as the τ segment of a morphism, f : T σ pMq Ñ A. Moreover these segments may be chosen freely because T σ pMq is the direct sum of the carapaces, M γ Òγ ad γ ranages over Gσ . Then, by equation 8.7, Gg ¨t fτ uτPGσ is the element of the same product whose τ-component is ΥGg,Gg´1 τ ˝ fGg´1 τ ˝ Gg´1 . This element of the product is G-stable if and only if ΥGg,σ ˝ fσ ˝ Gg´1 “ fGg σ . (9.5)

That is, if the family, t fτ u is G-invariant, each component is uniquely determined by fσ . Conversely, given fσ P HomGσ pM, Apσqq one may use Equation 9.5 to define fGg σ for eac, Gg, chosen that carries σ to Gg σ. Any other such element is of the form Gg h for some h P Gσ . But then replacing Gg by Gg h in 9.5) yields the same result because fσ is, by hypothesis, a Gσ -morphism. The proof for S σ pMq is too similar to bear repetition. Proposition 9.6. Let G act on X and let M be a representation of Gσ 121 over R. Then: ΣpX, T σ pMqq “ IG{Gσ pMq

ΓpX, S σ pMqq “ CG{Gσ pMq Proof. Begin by observing that for any R-module, N, ΣpX, N Òσ q “ N and ΓpX, Ndownarrowσ q “ N. Let U be some complete set of representatives of the cosets, Gg Gσ . Then, ž Σpx, T σ pMqq “ ΣpX, M γ Òγ q by Definition 9.1 “

“ “

ž

γPGσ

γPGσ

ž

ΣpX, M γ Òγ q

M xσ

xPU

ž xPU

bM “ IG{Gσ pMq

The corresponding compution for S σ pMq is: ź ΓpX, S σ pMqq “ ΓpX, Mγ Óγ q γPGσ

141

142

William J. Haboush “ “ “

ź xPU

ź

ΓpX, M xσ Ó xσ q M xσ

xPU

ź xPU

HomRrGσ s pxRrGσ s, Mq

“ HomRrGσ s p

ž xPU

xRrGσ s, Mq

“ HomRrGσ s pRrGsℓ , Mq “ CG{Gσ pMq . Proposition 9.7. Let G act excellently on X with retraction π : X Ñ Y Ď X. Then ž T σ pApσqq “ Apγq Òγ γPGσ

σ

S pApσqq “

ź

γPGσ

Apγq Óγ

š Proof. Let Φ be the G-structure on A. Then ś γPGσ Apγq admits a natural representation of G. If Gg P G let φGg “ γPGσ ΦGg,γ . It is understood that application of φg must be followed by reindexing š of components. Furthermore, the natural injection, jσ : Apσq Ñ γPGσ Apγq is Gσ equivariant. Thus, jσ extends to a G-morphism, jA : IG{Gσ Ñ Apγq. Then jA pGg bApσqq “ φGg p jA p1bApσqqq “ ΦGg,σ pApσqq “ ApGg σq. Since IG{Gσ pApσqq is a coproduct of the R-submodules, Gg bApσq, it is clear that jA is an isomorphism. Since Apσqγ “ Gg bApσq one sees 122 that jA restrict to isomorphisms jγ : Apσqγ Ñ Apγq which comprise an equivariant family in the sense that φGg ˝ jγ “ jGg γ ˝ hGg is left homothety by g. Now consider Apσqγ By definition it is HomRrGσ s pGg rrGσ s, Apσqq with Gσ -action, x f puq “ f px´1 ¨ uq “ f puxq and where Gg is any element carrying σ to γ. For some choice of Gg, let βγ p f q “ ΦGg,γ p f pGgqq. Choose x P Gσ . Then ΦGg x,γ f pGg xq “ ΦGg x,γ pX ´1 f qpGgq “ ΦGg,σ ˝ 142

Algebraic Representations of Reductive Groups over Local Fields 143 Φ x,σ pX ´1 f qpGgq “ ΦGg,σ f pGgq. Consquently, βγ is independent of the choice of Gg and the maps tβγ u are an equivariant family as above. To prove the two note ś that, by definiš statements of the proposition, σ γ γ S pApσqq “ γPG tion, T σ pApσqq “ γPGσ Apσqγ Òγ and š śσ Apσq Óγ . The isomorphisms in question, then, are γPGσ jγ Òγ and γPGσ βγ Ó . Proposition 9.8. Let pX, Gq be excellent with quotient, Y Ď X, and retraction, π. Let A be any G-carapace of R-modules on X. Then: š (1) T0 pAq “ σPY T σ pApσqq. ś (2) S0 pAq “ σPY S σ pApσqq. š (3) ΣpX, T0 pAqq “ σPY IG{Gσ pApσqq. That is ΣpX, T0 pAqq is a coproduct of induced modules. ś (4) ΓpX, S0 pAqq “ σPY CG{Gσ pApσqq. That is ΓpX, S0 pAqq is a product of coinduced modules. Proof. Statements 3) and 4) follow form 1) and 2) by a direct apllication of proposition 9.6 and so we need only prove 1) and 2). To prove 1) and 2), first notice that since G acts excellently, Ťwe may write the collection of simplices in X as the disjoint union σPY Gσ . Then,just apply Proposition 9.7. One obtains: ž T0 pAq “ Apτq τPX

ž ž

“

σPY τPGσ

“

σPY

ž

Apτq

T σ pApσqq

For calS 0 pAq, the proof is: S0 pAq “

by Proposition 9.7

ź ź

σPY τPGσ

143

Apτq

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William J. Haboush “

ź

σPY

S σ pApσqq

We leave the G-structures to the reader.

Before proceeding note that if Y Ď X, if σ P Y and if M is an R-module, we may perform the constructions M Òσ and M Óσ both 123 in X and in Y and that the results might differ. Rather than lingering upon unnecessary distinctions or unnecessarily complicating notation, we caution the reader to maintain a certain vigilance in reading the next proof. Lemma 9.9. Let G act excellently on X with retraction π : X Ñ Y and section, s. Then for any σ P Y, and Gσ -module, M, s˚ pT σ pMqq “ M Òσ . š Proof. By definition, T σ pMq “ γPGσ M γ Òγ . Evaluating, ž T σ pMqpλq “ Mγ . γPλXGσ

Suppose that two simplices, Gg σ and hσ both lie in λ. The action is excellent and so separated. It followed that Gg σ “ hσ. If λ P Y then Gσ X λ˜ is just one simplex and if σ P Y that simplex must be σ. That is, for any λ P Y, T σ pMqpλq “ M Òσ pλq which establishes the result. Suppose X is a simplicial complex and that M is a sheaf of groups on X. Let V be a carapace of R-modules on X. Suppose that for each σ P X, we are given a representation, ρσ : Mpσq Ñ AutR pVq. Then for each pair, σ Ď τ notice that Vpσq is naturally an Mpτq module simply by pulling back by the restriction, rσM,τ . Definition 9.10. Let X be a simplicial complex and let M be a sheaf of groups on X. Then a carapace of representations of M on X over R is a carapace of R-modules, V, together with a family of representations, ρσ : Mpσq Ñ AutR pVq such that for any pair, σ Ď τ, the expansion, eτV,σ , is an Mpτq-morphism. If A and B are two carapaces of representations of M on X over R, a morphism of carapaces of representations is morphism of carapaces which is an Mpσq-morphism for ecah simplex, σ. 144

Algebraic Representations of Reductive Groups over Local Fields 145 Notice that carapaces of representations of M are clearly an Abelian category. Generally, when there is no danger of confusion, we will just say M-carapace to denote a carapace of representations of M on X over R. Let G operate excellently on X with retraction, π : X Ñ Y. Write s : Y Ñ X for the inclusion. Since excellent actions are separated, Gσ “ Gˆ σ , is a sheaf of groups on X which we will write G˚ . If A is a G-carapace of R-modules on X, then A is certainly a carapace of representations over the sheaf of groups, G˚ . Finally notice that if f : X Ñ Y is a morphism of complexes and if V is a carapace of representations over the sheaf of groups, M on Y then it is purely formal to check that f ˚ pMq is a carapace of representations over the sheaf of groups, f ˚ M. Definition 9.11. Suppose that G operates excellently on X with retrac- 124 tion, π : X Ñ Y, and section, s : Y Ñ X. Let A be a G-carapace of R-modules on X. Then theprototype of X on Y is s˚ pAq. It is a carapace of representations over s˚G˚ , the restriction of the sheaf of stabilizers. It is patently obvious that s˚ is ana exact functor from the category of G-carapaces to the category of carapaces of representations of s˚G˚ on Y over R. More can be said. We will write G˚ for the restriction of the stabilizer sheaf to Y if there is no danger of confusion. Theorem 9.12. Let G act excellently on X with retraction π : X Ñ Y and section, s. Then s˚ is an isomorphism of categories from the category of G-carapaces of R-modules on X to the category of carapace of representations of s˚G˚ on Y over R. Proof. In this discussion the functors Ti and Ki on X as well as the corresponding functors associated to Y occur. Consequently we will use Ti and Ki exclusively for the functors associated to X. The corresponding functors on Y will be written T i and K i . Write CarR pG, Xq for the category of G carapaces of R-modules on X and CarR pG˚ , Yq for the category of carapaces of representations of G˚ in R-modules on Y. Then s˚ is an exact functor from CarR pG, Xq to CarR pG˚ , Yq. Define a functor from CarR pG˚ , Yq to CarR pG, Xq by the 145

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equation: T Y0 pAq “

ž

σPY

T σ pApσqq

(9.13)

ś š Then s˚ pT Y0 pAqq “ σPY s˚ pT σ pApσqqq “ σPY Apσq Òσ by Lemma 9.9. But then by the definition of the functor T 0 , this says that s˚ pT Y0 pAqq “ T 0 pAq. Let T Y1 pAq “ T Y0 pK 0 pAqq. For each σ P Y, there is a natural inclusion, K 0 pAqpσq Ñ T Y0 pAqpσq. By Proposition 9.3, 2), this inclusion š gives a unique map qσ : T σ pK 0 pAqpσqq Ñ T Y0 pAq. Let qA “ σPY qσ . Then qA maps T Y1 pAq to T Y0 pAq and its restriction to Y is just the natural map from T 1 pAq to T 0 pAq. Define a functor on CarR pG˚ , Yq to CarR pG, Xq by the equation: IY pAq “ CokerpqA q

(9.14)

We will show that IY is inverse to s˚ . First suppose that A is in CarR pG˚ , Yq. Then the sequence, T Y1 pAq ą qa ąą T Y0 pAq Ñ IY pAq Ñ 0 125

is exact. Apply the exact functor, s˚ . The result is a commutative diagram: s˚ T Y1 pAq T 1 pAq

qA

δ1

/ s˚ T 0 Y

/ T 0 pAq

πA

/ s˚ pIY pAqq

/0

/A

/0

By exactness of the rows and commutativity, s˚ IY pAq is isomorphic to A. Now consider B is CarR pG, Xq. First notice that proposition 3, (??) gives canonical maps, T σ pBpσqq Ñ B. Sum to obtain a canonical map, d0 T Y0 ps˚ Bq Ñ B Ñ 0. This map clearly vanishes on the image of T Y1 ps˚ Bq. Hence it induces a mapping, ξB : IY ps˚ Bq Ñ B. Apply the exact functor, s˚ and use what we have just proven. Then s˚ pIY pS ˚ Bqq “ s˚ pBq and ξB induces the identity on segments in Y. 146

Algebraic Representations of Reductive Groups over Local Fields 147 Now finally note that a G-morphism of G-carapaces which is an isomorphism on Y is an isomorphism. That is IY ps˚ pBqq » B functorially in B. It follows that IY is inverse to s˚ . In what follows, Theorem 9.12 will be a very essential and fundamental tool for analyzing CarR pG, Xq.

10 Recollections and Fundamentals; Buildings For the most part we follow the notation and conventions of [BTI] and [BT II]. Our purpose here is a brief review which will establish notation and emphasize one or two differences. Throughout K is a field complete with respect to the discrete rank one valuation, ω : K ˚ Ñ Z. Then O will be the center of ω, k will be the residue field of O and ξ : O Ñ k is the natural map. Let G be be a Chevalley group scheme defined over Z. Assume it to be split, simply connected, connected and of simple type. Let T be a maximal torus, let N be its Cartan subgroup and let B be a Borel subgroup containing T all given as group subschemes of G defined over Z. Each of these group schemes being a functor, applying any one of them to ξ gives a morphism which, in all cases, we will also call ξ from GpOq to Gpkq, BpOq to Bpkq, etc.. let G “ GpKq, let G0 “ Gp′q and let G “ Gpkq. Let B “ BpKq, let B “ Bpkq but let B “ tx P G0 : ξpxq P Bu. Let N “ NpKq, let T “ TK and let H “ N X B. Let X denote the finite free Z-module of characters of T, let Φ denote the roots of G with respect to T; let Φ` denote those positive with respect to B; let ∆ be a basis of simple roots in Φ` and let α˜ denote the largest root. Let Γ “ HomZ pX, Zq be the group of one parameter 126 subgroups of T and let Φ Ď Γ be the set of co-roots. Let tUα : α P Φu be a set of root subgroups let Uα “ Uα pKq and let xα : Uα Ñ Ga,Z be the natural isomorphism. Let Uα,n “ tu : ωpxα puqq ě nu and observe that Uα,0 “ UpOq. Let N0 “ N X G0 . Then Γ can be identified with the group scheme morphisms, γ : Gm,Z Ñ T. Write ă γ, χ ą for the value, γpχq when 147

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γ P Γ and χ P X. We may always think of γ as a map, map : K ˚ Ñ T and so we may write χpγpyqq “ yăγ,χą . Following this convention, we may define the action of N on Γ by the equation, pn γqpxq “ npγpxqqn´1 . For t P T write tχ for χptq. Then N acts on X by the equation χn ptq “ χpntn´1 q. Then N0 normalizes T and so acts on it by conjugation. Consider the semi-direct product, T Ÿ N0 . For any pt, nq P T trianglele f tN0 , define a mapping, τt,n : Γ P Γ by the equation, xτpt,nq pγq, χ ą“ ´ωptχ q` ăn γ, χy

Then τt,n is an affine transformation and pt, nq ÞÑ τpt,nq is an action of T Ÿ N0 on Γ. If n P T X N0 it is clear that τn,n´1 “ idΓ and so this action reduces to an affine action of T ¨ N0 “ N on Γ. If n P N, γ P Γ write n “ tn0 and let n γ “ τpt,n0 q pγq. It is straightforward to verify that if n γ “ γ for all γ P Γ then n P H. Let A “ Γ bZ R and extend the action of N to A by linearity. Let XR “ X bZ R and choose a form on XR invariant under the vector Weyl group. For λ, χ P XR write the form, pλ|χq. With this form, identify A 2α with XR and write αˇ “ pα|αq for each α P Φ. Since H acts trivially under τ, and since N{H is naturally isomorphic to the affine Weyl group τ induces an action of the affine Weyl group on A and it is a triviality that this is the canonical action. For any pair, pα, rq P Φ ˆ Z, let α˚ “ tx P A : αpxq ` r ě 0u. These are closed half spaces and they are in bijective correspondence with Φ ˆ Z. If the half space, α˚ corresponds to the pair pα, rq, write Uα˚ for the group, Uα,r defined above. Write Bα˚ . The closed half spaces, α˚ , are called the affine roots of G in A and we write Σ for the set of all affine roots of G in A. Define an equivalence on A by saying that x „ y if and only if x P α˚ if and only if y P α˚ for all α P Σ. These equivalence classes are the facets of A and they are the interiors of simplies (because G is of simple type). Their closures give a simplicial decomposition of A. The maximal dimensionalŤfacets are called chambers. They are the connected components of A α˚ PΣ Bα˚ . Let ∆˜ denote the set of affine roots pα, 0q, α P ∆ and p´β, 1q where β is the unique largest root relative to the dual Weyl chamber. Let S denote 148

Algebraic Representations of Reductive Groups over Local Fields 149 ˜ These 127 the set of reflections thorugh the hyperplanes, Bα˚ for all α˚ P ∆. reflections afford a Coxeter presentation of the affine Weyl group, N{H, so that (G, B, N, S) is a Tits system in G. Then N{H acts transitively on the chambers Ş of A. Let C 0 “ α˚ P∆˜ α˚ and let C0 be its interior. Let F be any contained in C 0 and let S F “ ts P S : spFq “ Fu. In pariticular S C0 “ H. For any F, let PF be the subgroup of G generate by B and some arbitararily chosen set of representatives of S F chosen and it is called the parahoric subgroup of G associated to F. By definition, the parahoric subgroup of G relative to the Tits system, pG, B, N, S q, are the conjugates of the subgroups, PF . We shall call the facets, F Ď C 0 the types of G with respect to the Tits system, pG, B, N, S q. If P is a parahoric subgroup of G, then P is conjugate to a unique group of the form PF for some F. Then we call F the type of P and we write F “ τP. Notice that our terminology differs slightly from [BTI] in which the subsets, S F are called the types. We may now describe I “ IpG, B, N, S q, the building of G with respect to the Tits system, pG, B, N, S q. As a point set, IpG, B, N, S q is the set of pairs, pP, xq where P parahoric and x P τpPq. Now G acts on this set by the equation, GgpP, xq “ pGg P Gg´1 , xq. Let x be any point of A. Then for some n P N, nx P C 0 . Then the point nx is uniquely determined. Let F be the smallest open facet containing nx. Let apxq “ pn´1 PF n, nxq. The group, n´1 PF n is uniquely determined just as nx is and so a maps bA into IpG, B, N, S q. (This map is called j in [BTI].) We may now make IpG, B, N, S q into a geometric simplicial complex. Its simplices are just G translates of closed facets in apAq and its vertices are translates of the special points. The G-translates of apAq are called the apartments of IpG, B, N, S q. There are several structures on I “ IpG, B, N, S q. First there is what is called an affine structure in [BTI]. If x and y are any two points in I they are contained in one apartment. Thus for any λ P r0, 1s, there is a point λx ` p1 ´ λqy determined by the apartment. This point is, however, independent of the apartment chosen and the operation which assigns to each pair, px, yq together with a real λ P r0, 1s, the point λx ` p1 ´ λqy is the affine structure. There is a G invariant metric whose 149

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restriction to any apartment is the metric is written dpx, yq. Finally there is a “bornology” on subsets of G. A set is called bounded if it is contained in a finite union of double cosets, BwB for w P N. Now let us record some statements particularly useful to us. They are for the most part simple rearrangements of statements in [BTI] and [BT II]. 128

Lemma 10.1. The action of G on IpG, B, N, S q is excellent with section, C0. Proof. First,m C 0 is a fundamental domain for the G action and G is transitive on maximal dimensional simplices. This implies that the action is separated. The coled chamber, C 0 , is isomorphic to the quotient if I by the G action and the inclusion of C 0 into I is clearly a section. Lemma 10.2. G acts transitively on the paris pF, Cq where F is a facet and C is an apartment containing F. Proof. This is just 2.26, p. 36 in [BTI].

Proposition 10.3. The action of B on IpG, B, N, S q is excellent with section. a : A Ñ I. Proof. The action of G on IpG, B, N, S q “ I is separated and so, a fortiori, the action of B is as well. Let C0 be the chamner associated ot B. Recall the definition of the retraction of I on A with center, C0 ([BTI] 2.3.5, P. 38). As we remarked, given any two facets, there is an apartment containing them. Thus for any fact, F Ď I, there is an apartment A1 containing F and C0 . By 10.2, there is an element Gg in G so that GgpC0 , A1 q “ pC0 , Aq whence Gg F Ď A. Let ρC0 ,A pF 1 q “ Gg F. We show that ρC´10 ,A pF 1 q “ B ¨ F. Suppose that ρC0 ,A pF 1 q “ F. Then, by definition, there is an apartment, A1 , containing C0 and F 1 and an element Gg P G, so that Gg A1 “ A, GgF 1 “ F and Gg C0 “ C0 . Since Gg C0 “ C0 , Gg P B and Gg F 1 “ F. Hence ρC´10 ,A pF 1 q Ď B ¨ F. The opposite inclusion is clear. To complete the proof that I{B “ A. we use, nearly unmodified, the proof of 2.3.2, p. 37 of [BTI]. Clearly, for any facet, F, the orbit, 150

Algebraic Representations of Reductive Groups over Local Fields 151 B ¨ F meets A. What remains to be shown is that B ¨ F contains exactly one facet in A. Suppose that F and F 1 are in A and that F “ bF 1 , b P B. Then, F “ nF0 , F 1 “ n1 F0 for some F0 Ď C 0 . The stabilizer of F (respectively, F 1 ) is nPF0 n´1 pn1 PF0 pn1 q´1 respectively). Then bn1 PF0 pn1 q´1 b´1 “ nPF0 n´1 . Since parahoric subgroups are self normalizing, n´1 bn1 P PF0 and hence, b´1 n P n1 PF0 . Let X “ S F0 and let W x be the subgroup of W generated by X. Then PF0 “ BWX B. By [?], IV, 6, Proposition 2, n1 PF0 Ď Bn1 WX B. If ν : N Ñ W is the natural surjection, this implies that νpnq P νpn1 qwX and so nPF0 n´1 “ n1 PF0 pn1 q´1 . But F is the fixed point set of nPF0 n´1 and F 1 is the fixed point set of n1 PF0 pn1 q´1 . Con- 129 sequently, since the two groups are equal, F “ F 1 . thus for any facet, B ¨ F contains exactly one facet in A. As this is true, in particular, for vertices, A is the quotient of I by the action of B and so the proposition is proven.

11 Mounmental Complexes In this section the term monumental is thought of as meaning resembling or having the scale of a building ad it is used in the history of art. We use it to describe certain G actions which have all the properties of the natural actions on buildings which are of interest to us. If X is a finite dimensional simplicial complex and if G is a group acting on it then a subcomplex, Y Ď X will be called homogeneous if, Whenever Y contains two simplices, τ and γ such that τ “ Gg γ for some Gg P G, there is an element s P G such that sY “ Y and τ “ sγ. Definition 11.1. Let G be a group and let X be a finite dimensional simplicial complex on which G is acting with a separared action. Then the action of G on X will be called monumental if and only if the following conditions hold: (1) Every simplex in X is contained in a maximal dimensional simplex. (2) G acts transitively on the maximal dimensional simplices. 151

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(3) For any simplex, σ in X the stabilizer, Gσ is self normalizing in G. (4) If σ is a maximal dimensional simplex, the stabilizer, Gσ , acts excellently and it admits a homogeneous section, Y Ď X. Suppose that G is a group acting monumentally on X. Then a maximal dimensional simplex will be called a closed chamber; its interior will be called a chamber. Let C be a closed chamber in X. Then separation of the action implies the for any simplex σ in X, the orbit, Gσ contains at most one simplex which is a face of C while homogeneity implies that there is always at least one. Consequently the action of G is excellent with quotient C and section equal to the inclusion of C in X. Let B “ bpCq denote the G stabilizer of C and let Y be a homogeneous subcomplex of X mapping isomorphically onto the quotient, X{B Then any translate of Y, Gg Y will be called an apartment of type Y. In general we will think of the type, Y as chosen and fixed once and for all and so we will often speak merely of apartments. 130 Let F be a field. Let H be a commutative Hopf algebra over F with comultiplication, muH : H Ñ H bF H, augmentation, ǫH : H Ñ F, and antipode, sH : H Ñ H. Then H will be called proalgebraic if it is reduced and a direct limit of sub-Hopf algebras finitely generated over F. Then S pecpHq is a proalgebraic group scheme over F. If H is proalgebraic over F let GH denote the group of F-valued points of H. That is GH is the group of F-homomorphisms from H to F. Let F pGH , Fq denote the ring of F functions on GH and let γH be the natural map from H to F pGH , Fq, namely γH paqpφq “ φpaq. We will say that H separates k-points if γH is injective. Definition 11.2. Let G be a group. A monumental G-complexis a simplicial complex, X, on which G is acting monumentally together with a G-carapace of commutative F-algebras with unit, pA, tΦGg uGgPG q, and G-morphisms, µA : A Ñ A bF A, sA : A Ñ A and ǫA Ñ F X so that that following conditions are satisfied: (1) Each of the morphisms, µA , sA and ǫA is a morphism of G-carapaces of commutative F-algebra and for any simplex, σ, Apσq, µA,σ , sA,σ , ǫA,σ is a profinite Hopf algebra over F. 152

Algebraic Representations of Reductive Groups over Local Fields 153 (2) For each pair of simplices, σ Ď τ, the expansion, eτA,σ is a surjective morphism of proalgebraic Hopf algebras. (3) The G structure, tΦGg uGgPG acts by isomorphisms of carapraces of proalgebraic Hopf algebras. That is ΦGg,σ is a Hopf algebra isomorphism for each Gg and σ. (4) For each σ let Gpσq “ GApσq . Then G is naturally a sheaf of groups with the G structure induced by Φ. Then Apσq is reduced for each σ and there is an isomorphism of G-sheaves, α : G˚ Ñ G. The first three conditions of Definition 11.2 are self explanatory but the third requires some amplification. First of all, if M is a simplicial sheaf on X and ψ is an automorphism of X, ψ˚ Mpσq “ Mpψpσqq. The G-structure on G˚ is that arising from conjugation. That is, define cGg,σ : Gσ Ñ GGg σ by the equation: cGg,σ pxq “ Gg x Gg´1

(11.3)

Now we explain the G-structure on G induced by Φ. The functor G is contravariant and so GpΦGg,σ q maps GpGg σq to Gpσq. Thus define a G-structure, tΓGg uGgPG by the equation: ΓGg,σ GpΦGg´1 ,Gg σ q

(11.4)

Recalling that elements of Gpσq are the F-homomorphisms from Apσq 131 to F, this map can be more explicitly written, ΓGg,σ pxq ´ x ˝ ΦGg´1 ,Gg σ . It is customary to write apxq for xpaq when a is in a ring and x is a F-point of the ring. We may use α to identify Gσ with Gpσq, writing, for a P Apσq and Gg P Gσ , apGgq to denote rασ pGgqspaq. With these conventions, Condition 3) is nothing more than the equation: rΦGg,σ paqspxq “ apGg´1 x Ggq

(11.5)

Henceforth of X is a monumental G-complex with carapace of FHopf algebras A we will just say that pG, X, Aq is a monumental Gcomplex over F. Further we will use α to identify G˚ with G and we 153

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will always view Apσq as a ring of function of Gσ . The definitions of this section contain all the properties of buildings which will be used lated on. The next section explains how the affine building of the group of K-valued points of some semi-simple group for some valued field, K satisfies all of these conditions for an appropriate choice of the carapace A.

12 The Main Examples In this section we will discuss three monumental complexes. The first two are quite straightforward by the third requires a short dicussion of some classical results of M. Greenberg. Such symbols as K, ω, k, O, G, G, B, B etc. mean just what they did in the previous section.

The Admissible Complex Let K be a locally compact, non-Archimedean field and let F be any (discrete) field. Let IpG, B, N, S q be the buildings associated to G. For each σI, Gσ , the parahoric subgroup associated to the facet, σ, is a profinite group. For each σ let A0F pσq denote the ring of locally constant F-valued functions on Gσ . For any set T , T F will denote the set of all F-valued functions on T . Then pGσ {MqF A0F pσq “ lim ÝÑ M

where M varies over the open normal subgroups of Gσ . Since each of the algebras, pGσ {MqF is in fact a finite dimensional Hopf algebra with augmentation and antipode and since the inclusionspGσ {M1 qF Ď pGσ {M2 qF when M2 Ď M1 , is a Hopf morphism, A0F pσq is a Hopf algebra with antipode and augmentation.It is clearly profinite and it is 132 also clear that the expansions are surjective Hopf morphisms. Let GF be 154

Algebraic Representations of Reductive Groups over Local Fields 155 the sheaf of F-points of the carapace A0F as in §11.2, 4). Then GF pσq is the set of algebra homomorphisms, F Homal pGσ {MqF q “ lim Homal F ppG σ {Mq , Fq F plim ÝÑ ÝÑ M

M

G {M “ lim ÝÑ σ M

“ Gσ That is, there is a canonical isomorphism of sheaves of groups, α : G˚ Ñ GF . Finally define the G structure tΦGg uGgPG by the equation, rΦGg,σ paqspxq “ apGg´1 x Ggq. Since I is the building of G 1),2) and 3) of §11 show that G acts monumentally on it. We have just observed that A0F and I satisfy Conditions 1) through 4) of Definition 10.2. Hence pI, A0F q is a monumental G complex. We shall call it the admissible complex of G over F. We note that the group. G, may be replaced by a ˜ central extension, G.

The Spherical Complex For this example we depart somewhat from usual terminology. Let F be an algebraically closed field and let G F “ GpFq be the group of F points of the Chevalley scheme, G which we assume to be of simple type. Construct a complex as follows. The vertices of S are the proper reduced maximal parabolic subgroup schemes of G F which we regard as the base extension of G to F. The set P1 , ldotsPn is a simplex in S if and only if the intersection P1 X . . . X Pn is parabolic. Let G F act on S by conjugation. For any σ P S, we write Gσ for the stabilizer. Then Gσ is just the intersection of the groups corresponding to the vertices of σ. A chamber is the set of maximal parabolics containing a maximal torus. We must first establish that the action of G on S is monumental. The first three conditions of Definition 11.1 are quite well known. The proof of the fourth condition is again a modification of the proof of 2.3.2 of [BTI]. Let B be a Borel subgroup containing the maximal torus, T , and let N be the normalizer of T . Let P be any parabolic subgroup of G. 155

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By [H], 6.1, the intersection of any two parabolic subgroups contains a maximal torus. Thus there is maximal torus, S , in P X B. Hence there is some element, b P B, so that bsb´1 “ T . Hence bPb´1 contains T . Let ApT q denote the apartment corresponding to T .m We have shown that for any facet, τ, the B orbit, Bτ meets ApT q. Now suppose that P and Q are two B-conjugate parabolics both of which contain T . Then there are element m and n in N and b in B and a parabolic P0 containing B so that P “ nP0 n´1 , Q “ mP0 m´1 and bPb´1 “ Q. Thus, bnP0 n´1 b´1 “ mP0 m´1 and so, since P0 is self 133 normalizing, bn P mP0 . By [Bo], IV, 2.5.2, bn P BMW0 B where W0 is the Weyl group of P0 . Hence nP0 n´1 “ mP0 m´1 , That is any two B conjugate parabolics containing T are necessarily equal. It follows that any B orbit Bτ meets ApT q in exactly one facet. Condition 4) of the definition in hence established. For any σ P S let Apσq be the coordinate ring of the parabolic subgroup, Gσ . It is an elementary exercise in the theory of algebraic groups to see that pG F , S, Aq is a monumental G F complex.

The Affine Complex To describe this complex, we must recall some classical results of M. Greenberg. Let k be a perfect field and let R be a ring scheme over ˜ k. Let X be any k scheme. Define a ringed space, RpXq as follows. Its topological space is the underlying topological space of X and we denote it bpXq. If U is open in bpXq, let GR,X pUq “ RpS pecpOX pUqqq. As this functor is representible, it is a sheaf of rings on bpXq. Call R a scheme of local rings if GR,X is a sheaf of local rings for each scheme, X and assume this to be the case. Let V “ Rpkq. Then for each X, GR,X is a sheaf of V algebras. Let ˜ RpXq “ pbpXq, GR,X q. Then R˜ is a covariant functor from k schemes to V local of S pecpVq schemes. Though we call the following the first theorem of Greenberg, it is not given as one theorem in [MGI] but it largely summarizes the content of §4 of that work, especially Propositions 1 to 4 of §4 and the extensions thereof in §6. 156

Algebraic Representations of Reductive Groups over Local Fields 157 12.1 First Theorem of Greenberg. Suppose the ring scheme, R, is a projective limit of schemes each isomorphic to affine n space over k for varying n. Then there is a right adjoint to the functor GR,X from the category of k schemes to the category of V-schemes. That is, there is a functor F so that for any k scheme, X, and any S pecpVq scheme, Y, the following holds: HomV pGR pXq, Yq “ Homk ´ pX, FpYqq Moreover F satisfies: (1) If Y is of finite type over S pecpVq, then FpYq is a projective limit of schemes of finite type over k. (2) If Y is affine then so is FpYq. (3) If Y is a group scheme over S pecpVq, then FpYq is group scheme over k in such a way that the adjointness isomorphism of (2) is a group morphism functorially in X. This brings us to what we will call the second theorem of Greenberg. 134 In this case we are assemblings parts of §6, Proposition 1 of [MGI] and Proposition 2 and the structure theorem of [MG II], §2. Assume that R “ lim where Rn is a ring scheme over k which is k. isomorphic to ÝÑ n

A nk , affine n ` 1 space over k. Let In be the scheme of ideals in R corresponding to the kernel of the projection R Ñ Rn . Let Inr be the scheme of ideals in Rr such that o Ñ Inr Ñ Rr Ñ 0 is an exact sequence of group schemes for the additive structure. Assume that Inr is affine n´r space over k and that Rr is a locally trivial fiber space over Rn with fibre InR and the it is in fact a vector bundle over Rn . Let Gn “ GRn and let Fn be its right adjoint. Let Un pYq be the kernel of FpYq Ñ Fn pYq and let Urn pYq be the kernel of Fr pYq Ñ Fn pYq, pr ą nq. 12.2 Second Theorem of Greenberg. Let Y be a smooth group scheme with connected fibres over S pecpVq. If r ě n ě 0, Urn is a finite dimensional unipotent group scheme. Moreover, Fr pYq is S pecpVq isomorphic to the total space of a vector bundale over F0 pYq. 157

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We will refer to bUn pYq (respectively, Urn pYq) as the congruence subscheme of bFpYq (respectively Fr pYqq of level n, and we shall call FpYq the realization of Y over k. Now we return to the notation and conventions, of §10. Assume that k in perfect. Let X “ IpG, B, N, S q. By Proposition 10.3, the G action os monumental. We will show that X is a monumental complex over G. One of the more astonishing results in [BT II] is that for any σ P X, there is group scheme over O, Mσ , so that the generic fiber of Mσ is the base extension to K of G and such that Mσ pOq “ Gσ . Since G is split and simply connected and ω is discrete, Mσ can be assumed to be smooth with connected fibers. These group schemes are determined up to isomorphism if one requires that they admit Bruhat decompositions of a particular type. (see section 4.6 of [BT II]) We may assume that the schemes Mσ are carried to each other by the conjugation action on the generic fibre. Finally we will assume that O is the ring of k points of a ring scheme which is a projective limit of affine spaces over k. This is the case when O is the ring of Witt vectors of k or the ring the of formal power series in one variable over k. Then for any σ P X, Gσ is the set of k points of FMσ , the Greenberg realization of Mσ over k. Now we may describe the affine complex. Choose O as above to be the k points of a ring scheme isomorphic to an inverse limit of affine spaces. Choose G, ω, B etc. as in §11 and let X “ IpG, B, N, S q. For each σinX, le Apσq be the k coordinate ring of FMσ . The verification that pG, X, Aq is a monumental complex is now an entirely routine af135 fair. This monumental complex over G is what we shall call the affine complex of G over k.

13 Locally Rational Carapaces In this section and for the remainder of this discussion pG, X, Aq will always denote a monumental G-complex over the field , k, Recall that the action of the G-structure on A can be described by the equation ΦGg,σ p f q “ f ˝cGg´1 where cy denotes conjugation by y. Recall also that if M is any proalgebraic group with coordinates ring, A, over k, and if V 158

Algebraic Representations of Reductive Groups over Local Fields 159 is a rational representation of M with structure map β : V Ñ Vbk A, then if M acts on A be conjugation, γGg p f q “ f ˝ cg´1 , then β is equivariant as a map from the representation, V, to the representation V bk A, where the actions on V and A are determined by β and γ respectively. Definition 13.1. Let pG, X, Aq be a monmental complex and let Φ be the G-structure on A. A locally rational carapace of representations on pG, X, Aq is a G-carapace of k-vector spaces on X, V, with G structure, ψ and a morphism of G-carapaces, beta : V Ñ V bk A so the for each σ, the map βσ : Vpσq Ñ Vpσq bk Apσq is a comodule structure map in such a way that for each Gg, ψGg,σ is morphism of comodules. A morphism of locally rational carapaces is a G-morphism of Gcarapaces, f : V Ñ U which is a comodule morphism on each segment. It is clear that the locally rational carpaces on pG, X, Aq are an abelian category. The reader is cautioned to note that that the G-structures on A and V exist quite apart from the local comodule structure map β. In particular A admits several local comodule structures corresponding to left translation, right translation and conjugation. These are non-isomorphic locally rational structures on the same G-carpace. The comultiplication µ : A Ñ A bk A may be regarded as the local structure corresponding locally to right translation. We will write Ar for A with this local comodule structure. The reader is cautioned to note that that the G-structures on A and V exist quite apart from the local comodule structure map, β. In particular A admits several local comodule structure corresponding to left translation, right translation and conjugation. These are non-isomorphic locally rational structures on the same G-carapace. The comultiplication, µ : A Ñ A bk A may be regarded as the local structure corresponding locally to right translation. We will write Ar for A with this local comodule structure. Proposition 13.2. Let pG, X, Aq be a monumental complex and let V, β be a locally rational carapace of representation on pG, X, Aq. Let mu, ǫ, s denote the structure morphisms on A. Then 159

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(1) The structures on A endow the segment, ΣpX, Aq, with the structure of a co-algebra with co-unit and antipode. (2) The structure morphism, β, makes the segment, ΣpX, Vq into a comodule over ΣpX, Aq. 136

(3) Both the co-algebra structure data on ΣpX, Aq and the comodule structure map on ΣpX, Vq are G morphisms of representations. Proof. Inn this proof, we will make use of two functional properties of the map, tA,B of Proposition 2.2, 3) which have not been established. Let A, A1 , B, B1 and C be k-carapaces on X and let α : A Ñ A1 and β : B Ñ B1 be morphisms. The two functorial properties, whose proofs we leave to the reader, are these: tA1 B1 ˝ Σpα b βq “ pΣα b Σβq ˝ tA,B

ptA,B b idΣC q ˝ tAbB,C “ pidΣA b tB,C q ˝ tA,BbC

(13.3)

Now define structure data on ΣA as follows. Let µΣ “ tA,A ˝ Σµ, let eσ “ Σe and let sΣ “ Σs. The proof, for example, of co-associativity is the following computation: pµΣ b idΣA q ˝ µΣ “ ptA,A b idΣA q ˝ ppΣµq b idΣA q ˝ tA,A ˝ Σµ “ ptA,A b idΣA q ˝ tAbA,A ˝ Σpµ b idA q ˝ Σµ

“ ptA,A b idΣA q ˝ tAb,A,A ˝ ΣpidA b µq ˝ Σµ by the co-associativity of µ “ pidΣA b tA,A q ˝ tAA,AbA ˝ ΣpidA b µq b Σµ

byp136q

“ pidΣA btA,A q ˝ pidΣA b Σµq ˝ tA,A ˝ Σµ “ ridΣA b ptA,A ˝ Σµqs ˝ ptA,A ˝ Σµq “ pidΣA b µΣ q ˝ µΣ

The proof of the remaining axioms making ΣA into a coalgebra with co-unit and antipode are similar and are, for that reason, left to the reader. 160

Algebraic Representations of Reductive Groups over Local Fields 161 Define the co-action on ΣV by the equation, βσ “ tV,A ˝ Σβ. The proof that this is co-associative in truly indentical to the above computation with the first A’s in the expression there replaced by V’s. The last numbered assertion just follows from functoriality. Definition 13.4. Let pG, X, Aq be monumental complex. The algebra of measures on pG, X, Aq, which we write L1 pG, X, Aq or, more briefly as L1 pXq when no confusion will result, is the algebra pΣAq˚ , the k-linear dual of ΣA. Much of what follows depends on a natural L1 pG, X, Aq-structure on the exoskeletal homology groups of a locally rational carapace. To construct this action we must examine the functor, T0 of §7, (1), the first term in the canonical brittle resolution. Recall that for any carapace š A, rT0 pAqspσq “ τĎσ Apτq and the expansions are the natural inclusions . Moreover, the boundary map, δ0 naturally maps T0 pAq into A. Proposition 13.5. Let pG, X, Aq be a monumental complex and let V be 137 a locally rational carapace on X with structure map β. Then (1) The carapace, T0 pVq, admits a natural structure map, β0 , making it into a locally rational carapace. (2) The structuremap , β0 is uniquely determined by the requirement that δ0 be a morphism of locally rational carapaces. (3) If Vpσq is finite dimensional for each σ then the same is true of T0 pVq. Proof. Suppose that pV, βq is locally rational and that τ Ď σ are two simplices in X. Then, pidVpτq b eσA,τ q ˝ βτ “ βτ,σ makes Vpτq inti a Apσq comodule in such a way that the diagram, Vpτq

βτ,σ

/ Vpτ b Apσqq eσ V,τ

V,T σ

Vpσq

βσ

/ Vpσq b Apσq

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š commutes. Thus pβ0 qσ “ τĎσ βτ,σ is comodule structure map on rT0 pVqspσq Then p˚q may be applied twice, once to prove that T0 pVq is a carapace of co-modules over A and again to prove that β0 commutes with the two comodule structure morphisms. One verifies directly that β0 is a G-morphism. The uniqueness statement is clear as is the finiteness statement. Proposition 13.6. Let pG, X, Aq be a monumental complex. Then for each rgeq0 the r’th exoskeletal cohomology, Hr pX, ´q, is covariant functor from the category of locally rational carapaces on X to the category of left L1 pG, X, Aq-modules. Proof. By Proposition 13.2, Σ is a functor from the category of locally rational carapaces to the category of ΣpAq co-modules. The identity functor takes ΣpAq-co-modules to L1 pXq “ pΣpAqq˚ -modules. Thus Σ is a covariant functor to the category of left ∞ pXq modules. By Proposition 13.5 and the definition of the canonical brittle resolution (Definition 7.1), the canonical brittle resolution of V is a resolution by locally rational carapaces with boundary maps which are morphisms of locally rational carapaces. Thus the Alexander chains are a complex of L1 pXq-modules. The result follows immediately. 138

We shall be working with subalgebras of L1 pXq. In consequence, a “working description” of it might be of use. First notice that L1 pXq “ Homk pΣA, kq “ HomX,k pA, kX q. Thus a typical element of L1 pXq is a family, tBσ uσPX where Bσ P Apσq˚ , the linear dual of Apσq. The coherence condition on the family, tBσ uσPX is the commutativity of: Bσ

Apσq

/k

eτA,σ

(13.7)

Apτq

Bτ

/k

for every pair, σ Ď τ. This should be understood in the following sense. For any pro-algebraic group, the linear dual of its coordinates ring is an algebra under convolution. If one group contains another as a closed 162

Algebraic Representations of Reductive Groups over Local Fields 163 subgroup, then the dual of its coordinate ring contains the dual of the coordinate ring of the subgroup. Thus, 13.7) Ş says that whenever σ Ď τ the Bσ P pApτqq˚ . In particular Bσ P γĚσ as γ ranges over the chambers containing σ. In each of the main examples, this means that Bσ is in linear dual of th coordinate ring of the radical of Gσ . This prompts usŞ to define the X-radical of a stabilizer, Gσ as the intersection, RGpσq “ γĚσ Gγ where the intersection is taken over all chambers, γ, which contain σ . Henceforth we will write A˚ pσq for the linear dual of Apσq. Recall that for any commutative Hopf algebra A, the dual algebra, A˚ can be identified with the algebra of k-linear endomorphisms of A which commute with left translation or alternatively with those that commute with right translation. That is, ω : A Ñ A is an endomorphism commuting with left translations, then φ “ eA ˝ ω P A˚ is an element of the dual such that φ ˚ a “ ωpaq for all a P A and where the operation of φ is by right convolution. There is a similar statement with left and right interchanged. With this in mind, it is clear that L1 pXq is the algebra of k endomorphisms of A which are co-module morphisms for left translation (but not necessarily G-morphisms). The G-action on L1 can now be described. If Gg P Gσ then, Gg can be thought of as a k-homomorphism form Apσq to k and so as an element of A˚ pσq and Gσ can be thought of as a subgroup of the unit group of A˚ pσq. Hence in this case Gg acts by true conjugation, Gg ¨B “ Gg B Gg´1 . More generally the action is pGg ¨Bqσ “ BGg σ ˝ ΦGg σ .

14 The Injective Co-generator 139

Let pG, X, Aq be a monumental complex. In this section we show that the category of locally rational carapaces admits an injective cogenerator. We give a particular injective cogenerator and we use it to construct a module category containing an image of the locally rational carapaces. Let C denote a chamber in X and let A denote an apartment containing C. The unadorned symbols, τi and Si will denote the canonical brittle and flabby resolutions on X (see 8.3) while TiC and SCi will denote 163

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those on C and TiA and SiA those on A. Let ι : C Ñ X be the inclusion. By Theorem 9.12, ι˚ is an isomorphism of categories from G-carapaces on X to G˚ -carapaces on C. Recall that IC denotes its inverse. The locally rational carapaces on X are a subcategory of the Gcarapaces on X ans so ι˚ carries them to a subcategory on C. The reader can work out their properties when necessary; we will frequently argue in that category rather than the category of locally rational carapaces on X. For a G-carapace on X, V, we will VC to denote its restriction to C. If R is the coordinate ring of the proalgebraic group, H, it admits several structures as a rational representation. Let µ, e and s be the structural data for R. Write Rℓ for the left translation module, Rτ for the right translations module and R unadorned for the conjugating action, γ x paqpgq “ apx´1 gxq. Notice that s establishes an isomorphism between Rℓ and Rτ . Consequently except when the particular features of a calculation or proof demand other wise we will always write Rτ for this representation. For a representation of H, M, write M 5 to mean the vector space, M refurbished with the trivial representation. If β : M Ñ M bk R is the coaction on M, then the coassociativity of β is precisely equivalent to the statement that β is an H morphism from M to M 5 bk Rτ It is also true that β is an H-morphism from M to M bk R where the tensor product is with respect to the given structure on M and the conjugating representation on R. If M is any vector space equipped with the trivial representation, then M bk Rτ is H-injective, (It is, in fact, co-free.) To prove it, let f : N Ñ Mbk Rτ be an H-map, and let j : N Ñ Q be an H-monomorphism. Let ξ : Q Ñ Q bk R be the coaction. Just choose φ : Q Ñ M so the φ˝ j “ pid M beq˝ f . Then it is straightforward to verify that pφbidR q˝ξ is a comodule map from Q to M bk R whose composition with j is just f. Suppose that P is a closed subgroup of H of finite codimension. If N is a representation of P then the induced algebraic representation is a representation of H, IH{P pNq together with a P morphism, 140 ǫN : IH{P pNq Ñ N inducing the Frobenius reciprocity isomorphism, HomH pW, IH{P pNqq “ HomP pW|P , Nq. To construct IH{P pNq consider 164

Algebraic Representations of Reductive Groups over Local Fields 165 N bk R. The subgroup P acts on this product diagonally through the given representation on N and right translation on R. In addition, H acts by left translation on R and the trivial action on N. The actions of H and P on N bk R commute and so pN bk RqP is an H sub-representation of N 5 bk Rℓ . Then, IH{P pNq “ pN bk RqP , and the map ǫN is the restriction of idN b e. Composing idN b s with the inclusion of IH{P pNq in N bk R, we obtain a functorial map: ı : IH{P pNq ãÑ N 5 bk Rτ

(14.1)

The following is crucial. Lemma 14.2. Let H be profinite with structural data as above. Let V be a rational representation of H. There is a vector space, U, and exact sequence, 0 Ñ V Ñ V bk Rτ Ñ U bk Rτ

Proof. Let α : V Ñ V 5 bk Rr be the coaction viewed as an H-morphism. We will construct a morphism ψ : V 5 bk Rτ Ñ pV bk Rq5 bk Rτ so that kerpψq “ impαq. Let nupv b aq “ v b 1 ˆ a and let pλ “ idV b m b idR q ˝ pα b idR b idR q ˝ idV rps b idR q ˝ µs. These are both H morphisms for notice that if τpv b r b sq “ v b s b r, then τ ˝ λ is a comodule structure on V 5 bk Rτ . With this interpretation, the kernel of τ ˝ pνλq is just the space of invariants for this action. But then V bk R is the module of sections for a homogeneous bundle on H and the image of α is the subspace of invariant sections. Applying τ again. we have shown that the kernel of ν ´ λ is the image. Notice that as a G˚ -carapace, AC is equipped with the conjugating representation. If V is a rational G˚ -carapace, then V bk Ac always denotes the tensor product with respect to this structure. The comodule structure, α : V Ñ V bk AC is a G˚ -morphism with respect to this structure. is not a G˚ -morphism with respect to the right translation action. Define the carapace, ACτ by the equation, calACr pσq “ Apσqτ . Let Ar “ IC pACr q. Clearly, A and Aτ are not isomorphic as G carapaces. Let SC0 pkc q “ IC . Since k is field, IC is injective. Let IACr “ IC bk ACr . Then idIc b µ makes IACτ into a locally rational carapaces of 165

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G˚ -modules. Tensoring the natural inclusion, kC ãÑ IC , with Aτ yields the natural inclusion: : ACτ Ñ IACτ 141

In general we will call a carapace locally finite if each of its segments is a finite module. Proposition 14.3. Let pG, X, Aq be monumental with chamber, C. Let J be an injective k-carapace on C. Then J bk ACr is injective in the category of rational G˚ -carapaces. Proof. The corresponding proof for a proalgebraic group globalizes. Let U and V be two rational G˚ carapaces with co-actions, α : U Ñ U bk AC and β : V Ñ V bk AC respectively. Let ν : U Ñ V be a G˚ -monomorphism and let f : U Ñ J bk ACr be any G˚ map. Let eA : A Ñ kC be the counit. Then pid J b eA q ˝ f maps U to the k injective, J. Hence there is a map, φ : V Ñ J such taht pid J beA q˝ f “ φ˝µ. Then pφ b idA q ˝ β maps V to J bk Ar and a routine computation shows it to be a G˚ -morphism extending f . Definition 14.4. Let pG, X, Aq be monumental with chamber, C. Let r σ be the σ Ď C be a face and let V be a rational Gσ -module. Let V carapace on C with values: r σ pτq “ IG {G pVq if τ Ď σ V τ σ σ r V pτq “ p0q otherwise

(14.5)

The expansions are the canonical maps corresponding to transitivity of r σ q will be written, induction. The locally rational carapace on X, IC pV || Vσ . Lemma 14.6. Let pG, X, calAq be monumental with chamber, C. Let V be a representation of Gσ for some face σ in C. Then if U is any locally rational G-carapace, || r σ q “ HomGσ pUpσq, Vq HomX,G pU, Vσ q “ HomC,G pι˚ U, V

166

Algebraic Representations of Reductive Groups over Local Fields 167 Proof. The first equality is just the isomorphism induced by the isomorphism of categories, ι˚ . To prove the secon, suppose the f P HomC,G˚ r σ q We must show that f is uniquely determined by fσ . If τ is not pιU, V r σ . If τ Ď σ, a face of σ then, fτ “ 0. Write eσV,τ for the expansion in V σ σ σ then eV,τ circ fτ “ fσ ˝ eU,τ . But eV,τ “ ǫV and so fτ is the Gτ -morphism uniquely determined by Frobenius reciprocity. The result follows. Lemma 14.7. Let pG, X, Aq be monumetal with chamaber, C. Then, for 142 each face, σ, Čτ σ “ pk Óσ q bk Aτ Apσq C

Proof. This is just the following observation. Let H be a group and let P be a closed subgroup with coordinate rings, R and S respectively. Then IH{P pS τ q “ Rτ and the identification is functorial. Theorem 14.8. Let pG, X, Aq be monumental with chamber, C. Then: ||

r σ (respectively, Vσ q is (1) If V is an injective rational Gσ -module then, V an injective rational G˚ carapace on C (respectively an injectively locally rational G-carapace on X). (2) The carapace, IX ACτ “ IC pIC q is injective in teh category of locally rational carapaces on X. (3) Let V be a locally rational carapace on X. Then there are k-vector spaces, M and N and an exact sequence of locally rational Gcarapaces: 0 Ñ V Ñ M bk IX Aτ Ñ N bk IX Aτ If V is locally finite, then M may be taken to be finite dimensional. Proof. To prove 1),notice that it suffices to prove the statement for rational G˚ carapces on C. Notice that Lemma 14.6 implices that the functor, r σ , carrying rational Gσ modules, V, to G˚ -carapaces is right adjoint to V the exact functor which associates to ca carapace its σ segment. The statement is an immediate consequence. 167

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To prove 2),notice that IX ACτ “ IC pIC qbk Aτ and so by Proposition 14.3 it is injective. Now we establish 3). Making use of the canonical equivalence, we may prove the correspondig statement for rational G˚ carapaces on C. Thus let V be rational on C. Let α : V Ñ V bk A be the coaction on V. For each face in C, Σ, we may use the correspondence of lemma 6 to construct the unique Ć σ , such that ψσ “ id. Now consider the canonical map, ψ : V Ñ Vpσq morphisms ıVpσq of (1). These morphisms induce a monomorphism of σ : Vpσq Ć σ Ñ rVpσq5 Óσ s bk Aτ . Let f σ “ ıσ ˝ ψ and G˚ -carapaces, ı ś let f “ σĎC f σ . This is an injective map from V to ź pVpσq5 Óσ bAτ q σĎC

143

š

5 5 Let M “ σĎC Vpσq and let jσ : Vpσq Ñ M be the natural inclusion. Let πσ be the natural projection of ź pVpσq5 Óσ bAτ q σĎC

š on Vpσq Óσ bAτ . Then IX σĎC pπσ ˝ fσ q is the required map. If V is locally finite, M is finite. To conclude the proof just note that this construction may be applied to the cokernel of the map we have just defined. Corollary 14.9. The carapace IX ACτ is an injective cogenerator for the category of locally rational G-carapaces on X. Proof. This is just Statement 3) of the theorem.

Corollary 14.10. Let EG,X “ EndX,G pIX Aτ q. Then the functor PX,B defined by PX,G pVq “ HomX,G pV, IX Aτ q is a fully faithfull contravariant embedding of the category of locally rational G-carpaces on X in the category of left EG,X modules. Proof. This follows from Corollary 14.9 by a direct application of the Gabriel Mitchell embedding theorem. 168

Algebraic Representations of Reductive Groups over Local Fields 169

15 Locally Rational Carapaces In this section we will construct a section from a certain subcategory of the category of EG,X modules to the category of locally finite locally rational carapaces. We begin with a preliminary description of EG,X . Recall first that is R is the coordinate ring of a proalgebraic group then thelinear endomorphisms of R commuting with right convolutions are just convolutions on the left with elements of the linear dual of R. If ω is such an involution, then let ω be the element of the dual defined by the equation, ωp f q “ rωp f qspeq. Then left convolution with ω is ω. Also notice that Frobenius reciprocity for the indentity subgroup is the equation, HomG pV, M b Rτ q “ Homk pV, Mq. Now observe that the carapace (on C), IACτ may be written as the ś Čτ σ . Hence, finite product, Apσq σĎC

HomX,G pIACτ , IACτ q “

ź

σ,τĎC

Čτ σ , Apσq Čσ σ q HomX , GpApσq

(15.1)

Čτ σ , Apσq Čτ τ “ p0q whenever Apσq Čτ σ pτq 144 By Lemma 14.6, HomX , GpApσq “ p0q. That is, the Hom is null unless σ Ě τ. When this condition does hold: τ τ Čτ σ , Apτq Čτ τ q “ Hom HomX,G pApσq Gpτq pApτq , Apτq q σ

Čτ “ Apτqτ . Thus the σ, τ component of the homomorbecause Apσq phism is an element B P Apτq˚ operating by left convolution. Hence an element of EG,X can be represented as a matrix, pBτ,σ qτĎσ , where Bτ,σ P pApτq˚ q. Notice also that if τ Ď σ then Apσq˚ Ď Apτq˚ . Notice also that if τ Ď σ then Apσq˚ Ď Apτq˚ . Consequently, if pBτ,σ qτĎσ and pB τ,σ qτĎσ correspond to two elements of EX,G , for any τ, γ, σ such thatτ Ď γ Ď σ, the product, Bτ,γ B γ,σ is defined and is an element of Apτq˚ . For a locally algebraic carapace, V, PX,G pVq may also be calucilated directly: ź PX,G pVq “ HomX,G pV, IAC q “ HomGpσq pVpσq, Apσqq σĎC

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˚ By the remarks above, Hom śGpσq pVpσq,˚ Apσqq “ Vpσq , the contragredient. Hence, PX,G pVq “ σĎc Vpσq . Moreover, if Bτ,σ is an element ´ ¯˚ of EX,G then Bτ,σ operates on eσV,τ puq for u P Vpσq˚ and peσv,τ q˚ the adjoint of the expansion in V. Having established this much, we leave the remainder of the proof of the following to the reader.

Proposition 15.2. Let pG, X, Aq be monumental and let V be locally rational on X. Then: (1) The ring EX,G is equal to the ring of matrices, pBτ,σ qτĎσ . The product of two such matirces is given by pBτ,σ q ¨ pB τ,σ q “ pδτ,σ q where ÿ δτ,σ “ Bτ,γ B γ,σ τĎγĎσ

(2) The module PX,G pVq is isomorphic as an additive group to the set of vectors pvσ qσĎC where the component, vσ is in the contragredient module, Vpσq˚ . The action of pBτ,σ q on pvσ q yields puσ q where ÿ γ uσ “ Bσ,γ pEV,σ q˚ pvγ q σĎĎC

For the remainder of this section, we will write I to denote the canonical co-generator, IACτ . Then, EX,G “ EndX,G pIq. First, restriction to the chamber, C, is an isomorphism to the category of G˚ -carapaces. On the category of locally algebraic G˚ -carapaces on C, the segment 145 over σ is an exact functor to the category of algebraic representations of Gpσq. Consequently, for each σ, the segment Ipσq evaluted as a carapace on C, is left EX,G -module and the action commutes with the co-action, Ipσq Ñ Ipσq b Apσq. Evaluating the restriction of I to C on the facet, σ, we obtain the module of vectors, paτ qτĚσ : aτ P Apσq, By the description of EX,G above the action of the marix, pBγ,λ qγĎλ , on the vector, paτ qτĄ . pBγ,λ q ¨ paτ qτĄσ “ pbτ qτĚσ

where bτ “ ΣCĚγĚτ Bτ,γ ¨ aγ 170

(15.3)

Algebraic Representations of Reductive Groups over Local Fields 171 Furthermore, since elements of EX,G act as morphisms of carapaces, the expansions, which are compositions of projection of functions, are left EX,G -morphisms. Notice that the result would have been quite different if we had evaluated before restricting of C. We write IC pσq to denote the EX,G -module, I|C pσq. Since X and G ar fixed throughout this section, we will write E for EX,G as long as no ambiguity will result. Ă be the GLemma 15.4. Let M be a finitely generated E-module. Let M carapace on X whose prototype on C has the σ-setment, HomE pM, IC pσqq Ă is a locally algebraic and the expansions, HomE pidM , eλI,γ q. Then M Ă is a G subcarapace of an finite direct carapace on X. Furthermore M sum of carapaces each isomorphic to I.

r to C. Being Proof. As usual we need only consider the restriction of M the composition of a convariant functor with a carapace, it is certainly a carapace. We propose to show that it is locally algebraic. Write α : IC pqσ Ñ IC pσq b Apσq be the coaction commuting with teh E action. Finite generation over E menas that there is an exact sequenece, E‘τ Ñ M Ñ p0q. Moreover tensoring with Apσq alos results in an exact sequence. Write alpha1 for the map from HomE pM, IC pσqq to HomE pM, Iσ b Apσqq and write α2 for the map indued when M is replaced by E‘τ . also notice that, for nay E-module, N, we may write HomE pE‘τ , N bk Uq “ pN bk Uqbτ “ HomE pE‘τ , Nq bk U for any k-vector space, U. Putting this all togethe, we obtain a commuttive diagram: 0

/ HomE pE‘τ , IC pσqq

/ HomE pM, IX pσqq

α2

α1

0

/ HomE pM, ICpσq b Apσqq

/ HomE pE‘τ , IC pσqq b Apσq

(15.5) First obseve that ther is a natural embedding on HomE pM, IC pσqq b 146 Apσq in HomE pM, IX pσqq b Apσqq. Then all that must be proven is that the image of α1 is contained in HomE pM, IC pσqq b Apσq for then α1 will a fortiori be coaction while the horizantal arrow in the first row of 15.5 is the embedding we establishing tha last assertion of the lemma. 171

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Let j denote the upper horizontal arrow in 15.5 and let j1 denote the lower one. Let taγ : γ P Γu be a basis for Apσq.řIf u P HomE pM, IC pσqb Apσqq, then we may write j1 is injective, u “ γ pφγ q|M b aγ . But this shows that each element of HomE pM, IC pσqbApσqq is in HomE pM, IC pσqq b Apσq. That is the two space are equal so the result is established. Definition 15.6. Let V be a locally algebraic G-carapace on X. We shall say that V in finitely congenerated if for some integer, r ě 0, there is an exact sequence, 0 Ñ V Ñ I‘τ . Since P x,G pIq “ E, and since I is injective, it is clear that PX,G carries finitely cogenerated carapaces to finitely generated E-modules. Ă carries finitely generated Further, by lemma 4, the functor M ÞÑ M E-modules to finitely cogenerated carapaces. Notice that finitely cogenrated carapaces are not an Abelian category. While subcarapaces of finitely cogenerated carapaces are finitely cogeneratd it is not clear that quotiendts of finitely cogenerated carapaces are finitely cogenerated. This is of courese dual to the question pof whether submodules of finitely generated E-modules are finitely generated. That is, it would imply left Noetherianness of E. Theorem 15.7. Let pG, X, Aq be monumetal. Then (1) The functor PX,G carries the category of finitely cogenerated carapaces on X to the category of finitely generated E-modules. Ă carries the category of finitely generated E(2) The functor M ÞÑ M modules to the category of finitely cogenerated carapaces on X.

(3) If V is locally finite carapace on X, there is anatural isomorphism from V to PX,G Ą pVq. Moreover these two functors are isomorphisms between the category of locally finite carapaces and the category of E- modules which are finite dimensional over k. Proof. The first statement in the theorem was proven in the paragraph above. The second statement follows immediately from Lemma 15.4. Only the last statemnet requires proof. 172

Algebraic Representations of Reductive Groups over Local Fields 173 147

š Since PX,G pVq “ σĎC Vpσq˚ , it is clear that when V is locally finite, calPX,G pVq is finite dimentsional. For the converse, let eλ,γ for λ Ď γ Ď C denote the element of E corresponding to the matrix whoseµ, ν ‰ pλ, γq and whose λ, γ entry is the identity in Apλq˚ . The elements eλ,λ are a complete set of othogonal idempotents. Further for each λ such that λ Ğ σ, eλ,λ IC pσq “ p0q. š If M is ˚anu left Emodule, write Mσ “ eσ,σ M Then, calM “ σĎC Vpσq , it is clear that when V is locally finite, PX,G pVq is finite dimensional. For the converse. let eλ,γ for λ Ď γ Ď C denote the element of E corresponding to the matrix whose µ, ν entry is 0 when pµ, νq ‰ pλ, γq adn whose λ, γ entry is the identity in Apλq˚ . The elements eλ,λ are a complete set of orthogonal idempotents, Further for each λ such that λ Ğ σ, eλ,λ IC pσq “ p0q. If M is any left E- module. write Mσ “ eσ,σ M then, M “ coprodγĎC Mτ . By simple restriction there is a map from HomE pM, IC pσqq “ HomApσq˚ pMσ , Apσqq. We propose to show that it is an isomorphism. By the remarks above, if f P HompM, IC pσqq, then f pMγ q “ p0q whenever γ Ğ σ. Hence it suffices to determine f on Mτ for τ Ď σ. If u “ eτ,τ u P Mτ then, f peσ,τ uq. But eσ,τ u P Mσ and so f is determined by its restriction to Mσ . Thus. HomE pM, IC pσqq “ HomApσq˚ pM, Apσqq

(15.8)

This has two consequences. First, for general M, there is a natural inclusion HomApσq˚ pMσ , Apσqq ãÑ M˚σ given by f ÞÑ e ˝ f where e just denotes evaluation at the identity. The image of this map is always algebraic and when Mσ is itself algebraic it is an isomorphism as remarked in the first paragraph of this section. Thus if M is finite diĂ mensional, the same can be said of Mpσq. Moreover, when V is locally ˚ algebraic, PX,G pVqσ “ Vpσq . Thus if V is locally finite, Vpσq˚ is finite dimensional and so the last part of Statement 3) follows at once.

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References [B1] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976) 233–259. [B2] A. Borel, Linear Algebraic Groups, second enlarger edition, Springer-Verlag, New York (1991). [BW] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups and Representations of Reductive Groups, Annals of Mathematics Studies, 94 Princeton University Press. Princeton, New Jersey (1980) 3–381. 148

´ ements de math´ematique, Fasc. XXXIV, Groupes [Bo] N, Bourbaki, El´ et Alg`ebres de Lie, IV, V and VI, Actualit´es Scientifiques et Industrielles, 1337 Hermann, Paris (1968). [Br] K.S. Brown, Buildings, Springer-Verlag, New York (1989) 1–209. ´ [BTI] F. Bruhat and J. Tits, Groupes R´eductifs sur un Corps Local I. Donn´ees radicielles valu´ees Publ. Math IHES, 41 (1972) 5–252. [BT II] F. Bruhat and J. Tits, Groupes R´eductifs sur un Corps Local II. Sch´emas en groupes. Existence d’une don´ee radicielle valu´ee, Publ. Math. IHES, 60 (1984) 5–184. [CE] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, New Jersey (1956). [C] P. Cartier, Representations of p-adic groups, A survey in Automprphice forms, representatons and L-functions I, Proceeding on the Symposium in pure Mathematics, 33 American Mathematical Society, Providence, Rhode Island (1979) 111–155. [CW] W. Casselman and D. Wigner, Continuous cohomology and a conjecture of Serre’s, Invent. Math 25 (1974) 199–211. 174

Algebraic Representations of Reductive Groups over Local Fields 175 [Go] R.Godement, Th´eorie des faisecaux, Act. Sci. Ind. 1252 Hermann, Paris (1958) [MGI] M. J. Greenberg, Schemata over local rings, Ann. Math 73 (1961) 624–648. [MG II] M . J. Greenberg, Schemata over local rigns II, Ann. Math. 78 (1963) 256–266. [Gr] A. Grothendieck Sur quelques points d’alg`ebrr homologiguw, Tohoku Math. Jour. 9 (1957) 119–221. [H] W. J.Haboush, Reductive groups are geometrically reductive, Ann. Math. 102(1975) 67–83. [Hu] J. E. Humphreys. Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York(1991) [NM] N. Iwaori and H. Matsumoto, On some Bruhat decomposiotion and teh structre of the Hecke rings of p-adic Chevalley groups, Publ. Math. IHES 25 (1965) 5–48. [KL] D. Kazxhadan and G. Lausztig, Proof of the Deligne-Langlands 149 conjecture for Hecke algebras, Inv. Math. 87 (1987) 153–215. [L1] G. Lusztig, Hecke algebras and Jangtzene’s generic decompsition paterns, Adv. in Math. 37 (1980) 121–164. [L2] G. Lusztig,Some examplex of squar integrable representations of p-adic groups, Trans. Amer. Math. Soc. 277(1983) 623–653. [L3] G. Lusztig, Equivariant K-theory and representations of Hecke algebras, Proc. Amer. Math. Soc. 94 (1985) 337–342. [L4] G. Lusztig, Equivariant K-theory and representations of Hecke algebras II, INv. Math 80 (1985) 209–231 [M] I. G. Macdonald, Spherical fucntions on a groups of p-adic type, Ramanujan Institute Lecture Notes, Ramanujan Institute, Centre for Advanaced Study in Mathematics, Madras (1971) 1–79. 175

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[Mac] S. MacLane. Homology, Die Grundlehren der Mathematischen Wiesensaften in Einzeldarstellungen Vol. 114, Academic Press, Springer-Verlag, Berlin-G¨ottingenr-Heidelberg (1963) 1–422. [PS] A. pressely and G. Segal, Loop Groups, Clarendon Press, Oxford (1986). [R] M. Ronan, Lectures on buildings, Perspectives in Mathematics, 7 Academic Press, Harcourt Brace Jovanovich, Boston-San DiegoNew York-Derkeley-London-Sydney-Tokyo-Toronto (1989) 1– 201. [RS] M. Ronan and S. d. Smith, Sheaves on buildings and modular representations of Chevalley groups, J. Algebra 96 (1985) [S] L. Solomon, Mackey formula in the group ring of a Coxeter group, J. Aglebra 41 (1976) 225–264. [T] J. Tits, Two properties of Coxeter complexes, appendix to A Makey formula in the group in the group ring of a Coxerter group J. Algebra 41 (1976) 255–264.

Department of Mathematics University of Illinois 273 Altgeld Hall 1409 W. Green St. Urbana, Illinois 61801 U.S.A.

176

Poncelet Polygons and the Painleve´ Equations N. J. Hitchin Dedicated to M.S. Narasimhan and C. S. Seshadri on the occasion of their 60th birthdays

1 Introduction The celebrated theorem of Narasimhan and Seshadri [13] relating stable 151 vector bundles on a curve to unitary representations of its fundamental group has been the model for an enormous range of recent results intertwining algebraic geometry and topology. The object which meditates between the two areas geometry and topology. The object which mediates between the two areas in all of these generalizations is the notion of a connection, and existence Theorems for various types of connection provide the means of establishing the theorems. In one sense, the motivation for this paper is to pass beyond the existence and demand more explicitness. What do the connections look like ? Can we write them down? This question is our point of departure. The novelty of our presentation here is that the answer involves a journey which takes us backwards in time over two hundred years form the proof of Narasimhan and Seshadri,s theorem in 1965. For simplicity, instead of considering stable bundles on curves of higher genus we consider the analogous case of parabolically stable bundles, in the sense of Mehta and Seshadri [11], on the complex projective line CP1 . Such a bundle consists of a vector bundle with a weighted flag structure at n marked points a1 , . . . , an . The unitary connection that is associated with it is flat and has singularities at the points. In the generic case, the vector bundle itself is trivial, and the flat connection we are looking for can be written as a meromorphic mˆm matrix-valued 1-form with a simple pole at each point ai . The parabolic structure can easily be read off form the residuces of the form. The other side of the equation is a representation of the fundamental group π1 pCP1 ta1 , ldots, an uq in 177

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Upmq. the holonomy of the connection, and this presents more problems. Such questions occupied the attention of Fuchs, K-lein and others 152 in the last century under the alternative name of monodromy of ordinary differential equations. Now if we fix the holonomy, and ask for the corresponding 1-form for each set of distinct points ta1 , . . . an u Ă CP1 , what in fact we are asking for is a solution of a differential equation, the so-called Schlesinger equation (1912) of isomonodromic deformation theory. To focus things even more, in the simple case where m “ 2 and n “ 4, and explicit form for the connection demands a knowledge of solutions to a single nonlinear second order differential equation. This equation, originally found in the context of isomonodromic deformations by R. Fuchs in 1907 [4], is nowadays called Painlev´e’s 6th equation ˙ ˆ ˙2 ˆ ˙ ˆ dy dy 1 1 1 1 1 1 d2 y ´ ` ` ` ` “ 1{2 2 y y´1 y´x dx x x ´ 1 y ´ x dx dx ˆ ˙ ypy ´ 1qpy ´ xq xpx ´ 1q x x´1 ` α ` β ` γ ` δ x2 px ´ 1q2 y2 py ´ 1q2 py ´ xq2

and in the words of Painlev´e, the general solutions of this equation are “transcendantes essentiellement nouvelles” That, on the face of it, would seem to be the end of the quest for explicitness-we are faced with the insuperable obstacle of Painlev´e transcendants. Notwithstanding Painlev´e’s statement, for certain values of the constants α, β, γ, δ, there do exist solutions to the equation which can be written down, and even solutions that are algebraic. One property of any solution to the above equation is that ypxq can only have branch points at x “ 0, 1, , 8. This is essentially the “Painlev´e property”, that there are no movable singularities. If we find an algebraic solution, then this means we have an algebraic curve with a map to CP1 with only three critical values. Such a curve has a number of special properties. On the one hand, it is defined by a subgroup of finite index in Γp2q Ă S Lp2, Zq, and also,by a well-known theorem of Weil,is defined over Q. In this paper we shal construct solutions by considering the case when the holonomy group Γ of the connection is finite. In that case the solution ypxq to the Painlev´e equation is algebraic.

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Our approach here is to consider, for a finite subgroup Γ of S LpX, Cq, the quotient space S Lp2, Cq{Γ and an equivariant compactification Z. Thus Z is a smooth projective threefold with an action of S Lp2, Cq and a dense open orbit. The Maurer-Cartan form defines a flat connection on S Lp2, Cq{Γ with holonomy Γ, which extends to a meromophic connection on Z. The idea is then to look for rational curves in Z such taht the induced connection is of the required form. By construction the holonomy is Γ, and if we can find enough curves to vary the cross-ratio of the singular points a1 , . . . a4 , then we have a solution to the Painlev´e equation. The question of finding and classifying such equivarian compactifications has been addressed by Umemura and Mukai [12], but here we 153 focus on one particular case. We take Γ to be the binary dihedral group D˜ k Ă S Up2q. This might seem very restrictive within the context of parabolically stable bundales, but behind it there hides a very rich seam of algebraic geometry which has its origins further back in history than Painlev´e. In the case of the dihedral group, the construction of a suitable compactification is due to Schwarzenberger [16], who constructed a family of rank 2 vector bundles Vk over CP2 . THe threefold corresponding to the dihedral group Dk turns out to be the projectivizesd bundle PpVk q. There are two types of relevant rational curves. Those which project to a line in CP2 yield the solution y “ sqrtx to the Painlev´e equation with coefficients pα, beta, γ, δq “ p1{8, ´1{8, 1{2k2 , 1{2 ´ 1{2k2 q. Those which project to a conic lead naturally to another problem, and this one goes back at least to 1746 (see [3]). It is the problem of Poncelet polygons. We seek conics B and C in the plane such that there is a k-sided polygon inscribed in C and circumscribed about B. Interest in this problem is still widespread. Poncelet polygons occur in questions of stable bundales on projectives spaces[14] and more recently in the workl of Barth and Michel [1]. In fact, we can use their approach to find the modular curve giving the algebraic solution ypxq of the Painlev´e equation corresponding to Γ “ D˜ k . This satisfies Painlev´e equation with coefficients pα, β, γ, δq “ p1{8, ´1{8, 1{8, 3{8q. It is essentially Cayley’s solution in 1853 of the Poncelet problem which allows us to go further and produce explicit solutions. It is method which fits in well 179

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with the isomonodromic approach. There are a number of reasons why this is a fruitful area of study. One of them concerns solutions of Painlev´e equation in general and their relation to integrable systems, another is the connection with self-dual Einstein metrics as discussed in [6]. In the latter context, the threefolds constructed are essentially twistor spaces, and the rational curves twistor lines,but we shall not pursue this line of approach here. Perhaps the most intriguing challenge is to find any explicit solution to an equation to which Painlev´e remark refers. The structure of the paper is as follows. In Section 2 we consider singular connections and the isomonodromic deformation problem, and in Section 3 see how equivariant compactifications give solutions to the problem. In Section 4 we look at the way the dihedral group fits in with the problem of Poncelet polygons. Section 5 and 6 discuss the actual solutions of the Painlev´e equation, especially for small values of k. Only there can we see in full explicitness the connection which, in the context of the theorem of Narasimhan and Seshadri, relates the parabolic struc154 ture and the representation of the fundamental group, however restricted this example may be. In the final section we discuss the modular curve which describes the solutions so constructed. The author wishes to thank M.F. Atiyah and A. Beauville for useful conversations.

2 Singular connections We intrduce here the basic objects of our study -flat meromorphic connections with singularities of a specified type. For the most part we follow the exposition of Malgrange [10]. Definition 1. Let Z be a complex manifold, Y a smooth hypersurface and E a holomorphic vector bundle over Z. let ∇ be a flat holomorphic connection on E over Z Y with connection form A in some local trivialization of E. Then on U Ď Z we say that (1) ∇ is meromorphic if A is meromorphic on U. 180

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(2) ∇ has a logarithmic singularity along Y if, in a local coordinate system pz1 . . . , zn q of Z, with Y given by z1 “ 0, A has the form A “ A1

dz1 ` A2 dz2 ` . . . ` An dzn z1

where Ai is holomorphic on U. One may easily check that the definition is independent of the choice od coordinates and local trivialization. The essential point about a logarithmic singularity is that the pole only occurs in the conormal direction to Y. In fact ∇ defines a holomorphic connection on E restricted to Y, with connection form AY “

n ÿ

i“2

Ai p0, z2 , . . . , zn qdzi .

If Z is 1-dimensional, then such a connection is just a meromorphic connection with simple poles. Flatness is automatic because the holomorphic curvature is a (2,0) form which is identically zero in one dimension. If we take Z “ CP1 , Y “ ta1 , . . . , an , 8u and the bundle E to be trivial, then A is a matrix-valued meromorphic 1-form with simple poles at z “ a1 , . . . , an , 8 and can thus be written as i ÿ Ai dz A“ z ´ ai i“1

The holonomy of a flat connection on ZzY is obtained by parallel trans- 155 lation around closed paths and defines, after fixing a base point b,a representation of the fundamental group ρ : π1 pZzYq Ñ GLpm, Cq In one dimension, the holonomy may also be considered as the effect of analytic continuation of solutions to the system of ordinary differential equations n ÿ df Ai f ` “0 dz i“1 z ´ ai 181

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around closed paths through b. As such, one often uses the classical term monodromy rather than the differential geometric holonomy. Changing the basepoint to b1 effects an overall conjugation (by the holonomy along a path from b to b1 ) of the holonomy representation. For the punctured projective line above, we obtain a representation of the group π1 pS 2 zta1 , . . . , an , 8uq. This is a free group on n generators, which can be taken as simple loops γi from b passing once around ai . Moving b close to ai , it is easy to see that ρpγi q is conjugate to expp´2πiAi q. There is also a singularity of A at infinity with residue A8 . Since the sum of the residues of a differential is zero, we must have A8 “ ´

n ÿ

Ai

i“1

and so ρpγin f ty q is conjugate also to expp´2πiA8 q. In the fundamental group itself γ1 γ2 . . . γn γ8 “ 1 so that in the holonomy representation ρpγ1 qρpγ2 q . . . ρpγ8 q

(1)

Thus the conjugacy classes of the residuces Ai of the connection determine the conjugacy classed of ρpγi q, and these must also satisfy (1). This is partial information about the holonomy representation. However, the full holonomy group depends on the position of the poles ai . The problem of particular interset to us here is the isomonodromic deformation problem to determine the dependence of Ai on a1 , . . . , an in order that the holonomy representation should remain the same up to conjugation. All we have seen so far is that the conjugacy class of expp2πiAi q should remain constant. 156 One way of approaching the isomonodromic deformation problem, due to Malgrange, is via a universal deformation space. Let Xn denote the space of ordered distinct points pa1 , . . . , an q P C, and X˜ n its universal covering. It is well-known that this is a contractible space- the 182

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183

classifying space for the braid group on n strands. Now consider the divisor Ym “ tpz, a1 , . . . , an q P CP1 ˆ Xn : z ‰ am u and let Y˜ m be its inverse image in CP1 ˆ X˜ n . Furthermore, define Y˜ 8 “ tp8, xq : x P X˜ n u. The projection onto the second factor p : CP1 zta1 , . . . , an , 8u, and the contractibility of X˜ n implies fro the exact homotopy sequence that the inclusion i of a fibre induces an isomorphism of fundamental groups π1 pCP1 zta01 , . . . , a0n , 8uq – π1 pCP1 ˆ X˜ n ztY˜ 1 Y . . . Y Y˜ n Y Y˜ 8 uq Thus a flat connection on CP1 zta01 , . . . , a0n , 8u extends to flat connection with the same holonomy on CP1 ˆ X˜ n ztY˜ 1 Y. . .Y Y˜ n Y Y˜ 8 u. Malgrange’s theorem asserts that this flat connection has logarithmic singularities along Y˜ m and Y˜ 8 . More precisely, Theorem 1 (Malgrange [10]). Let ∇0 be flat holomorphic connection on the vector bundle E 0 over CP1 zta01 , . . . , a0n , 8u, with logarithmic singularities at a01 , . . . , a0n . Then there exists a holomorphic vector bundle E on CP1 ˆ X˜ n with a flat connection ∇ with logarithmic singularities at Y˜ 1 , . . . , Y˜ n , Y˜ 8 and an isomorphism j : i˚ pE, ∇q Ñ pE 0 , ∇0 q. Furthermore, pE, ∇, jq is unique up to isomorphism.

Now suppose that E 0 is holomorphically trivial. The vector bundle E will not necessarily be trivial on all fibres of the projection p, but for a dense open set U Ď X˜ n it will be. Choose a basis e01 , e02 , . . . e0m of the fibre of E 0 at z “ 8. Now since ∇ has a logarithmic singularity on Y˜ 8 , it induces a flat connection there, and since Y˜ 8 – X˜ n is simply connected, by parallel translation we can unambiguously extend e01 , e02 , . . . , e0m to trivialization of E over Y˜ 8 . Then since E is holomorphically trivial on each fibre over U. we can uniquely extend e01 , e02 , . . . , e0m along the fibres to obtain a trivialization e1 , . . . , em of E on CP1 ˆU. It is easy to see that, relative to this trivialization, the connection form pf ∇ can be written A“

n ÿ

i“1

Ai

dz ´ dai z ´ ai

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where Ai is a holomorphic function of a1 , . . . , an . The flatness of the connection can then be expressed as: dAi `

ÿ

rAi , A j s

j‰i

dai ´ da j “0 ai ´ a j

which is known as Schlesinger’s equation [15]. The gauge freedom in this equation involves only the choice of the initial basis e01 , e02 , . . . , e0m and consists therefore of conjugation of the Ai by a constant matrix. The case which interests us here in where the holonomy lies in S Lp2, Cq, (so that the Ai are trace-free 2 ˆ 2 matrices), and where there are 3 marked points a1 , a2 , a3 which, together with z “ 8, are the singular points of the connection. By a projective transformation we can make these points 0,1, x, Then Apzq “

A1 A2 A3 ` ` z z´1 z´x

and Schlesinger’s equation becomes:

rA3 , A1 s dA1 “ dx x rA3 , A2 s dA2 “ dx x´1 ´rA3 , A1 s A3 , A2 dA3 “ ´ dx x x´1

(3)

where the last equation is equivalent to

A1 ` A2 ` A3 “ ´A8 “ const. The relationship with the Painlev´e equation can best be seen by following [8]. Each entry of the matrix Ai j pzq is of the form qpzq{zpz ´ 1qpz ´ xq for some quadratic polynomial q. Suppose that A8 is diagonalizable, and choose a basis such that ˆ ˙ λ 0 A8 “ 0 ´λ 184

Poncelet Polygons and the Painlev´e Equations

185

then A12 can be written A12 pzq “

kpz ´ yq zpz ´ 1qpz ´ xq

(4)

for some y P CP1 zt0, 1, x, 8u. If the Ai pxq satisfy (3), then the function 158 ypxq satisfies the Painlev´e equation ˆ ˙ ˆ ˙2 ˆ ˙ 1 d2 y dy dy 1 1 1 1 1 “ 1{2 ` ` ` ` ´ 2 y y´1 y´x dx x x ´ 1 y ´ x dx dx ˆ ˙ xpx ´ 1q ypy ´ 1qpy ´ xq x x´1 α`β 2 `γ (5) `δ ` x2 px ´ 1q2 y py ´ 1q2 py ´ xq2

where α “ p2λ ´ 1q2 {2 β “ 2d det A21

γ “ ´2 det A22

δ “ p1 ` 4 det A23 q{2

(6)

For the formulae which reconstruct the connection from ypxq we refer to [8], but essentially the entires of tha Ai are rational functions of x, y and dy{dx. For our purposes it is useful to note the geometrical form of the definition of ypxq given by 4: Proposition 1. The solution ypxq to the Painlev´e equation corresponding to an isomonodromic deformation Apzq is the point y P CP1 zt0, 1, x, 8u at which Apyq and A8 have a common eigenvector. Note that strictly speaking there are two Painlev´e equations (with α “ p˘2λ ´ 1q2 {2q correspinding to the values of y with this property.

3 Equivariant compactifications Consider the three- dimensional complex Lie group S Lp2, Cq and the Lie algebra-valued 1-form A “ ´pdgqa´1 . 185

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The form A is the connection form for a trivial connection on the trivial bundle. It simply relates teh trivializations of the principle frame bundle by left and right translation. Now let Γ be a finite subgroup of S Lp2, Cq. Then S Lp2, Cq{Γ is non-compact ciompelx manifold and since A is invariant under right translations,it descentds to this quotient. Thus, on S Lp2, Cq{Γ, A defines a flat connection on the trivial rank 2 vector bundle. Its holonomy is tatutologically Γ. In this section, we shall consider an equivarient compactification of S Lp2, Cq{Γ, that is to say, a compact compelx manifold Z on which 159 S Lp2, Cq acts with a dense open orbit with stabilizer conjugate to Γ. Let Z be such a compactification, then the action of the group embeds the lie algebra g in th space of holomorphic vector fields on Z. Equivalently, we have a vector bundle homomorphism α : Z ˆ g Ñ TZ which is generically an isomorphism.It fails to be an isomorphism on the union of the lowerŹ dimensional orbits of S Lp2, Cq, and this is where Ź Ź 3 α P H 0 pZ, Homp 3 g, 3 T qq – H 0 pZ, K ´1 q is a section of the anticanonical bundle, so the union of the orbits of dimension less than three form an anticanonical divisor Y, which may of course have several components or be singular. In the open orbit Z{Y – S Lp2, Cq{Γ, the action is equivalent to left multiplication, and the connection A above is given by A “ α´1 : T Z Ñ Z ˆ g. It is clearly meromorphic on Z, but more is true. Ź Proposition 2. If 3 α vanishes non-degenerately on the divisor Y, then the connection A “ α´1 has a logarithmic singularity along Y. This is a local statement, and so it can always be applied to the smooth part of Y even if there are singular points. Proof. In local coordinates, α is represented by a holomorphic function Bpzq with values in the space of 3 ˆ 3 matrices. The divisor Y is then 186

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187

the zero set of det B. If det B has a non-degenerate zero at p P Y, then its null-space is one-dimensional at p, so the kernel of α, the the Lie algebra of the stabilizer of p, is one-dimensional. Thus the S Lp2, Cq orbit through p is two-dimensional, and so Y is the orbit. Now for any quare matrix B, let B_ denote the transpose of the matrix of cofactors. Then it is well-known that BB_ “ pdet BqI Hence in local coordinates A “ α´1 “

B_ det B

and so A has a simple pole along Y. From Definition 1, we need to show 160 that the residue in the conormal direction. For this consider the invariant description of B_ . We have on Z ^2 α : ^2 g Ñ ^2 T and using the identifications ^2 g – g˚ and ^2 T – T ˚ b ^3 T, B_ represents the dual map of ^2 α: p^2 αq˚ : T Ñ g b ^T. Now the image of α at p is the tangent space to the orbit Y at p by the definition of α. Thus the image of ^2 α is ^2 T Y p which means that p^2 αq˚ annihilates T Y, which is the required result. Note that the kernel of ^2 α is the set of two-vectors v ^ w where w P g and v is ion the Lie algebra of the stabilizer of p. Thus the residue at p of the connection A lies in the Lie algebra of the stabilizer. Now suppose that P is a rational curve in Z which meets Y transversally at four points. Then the restriction of A to P defines a connection with logarithmic poles at the points and, from the map of fundamental groups π1 pPzta1 , . . . , a4 uq Ñ π1 pZzYq Ñ Γ Ñ S Lp2, Cq, 187

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its holonomy is contained in Γ. A deformation of P will define a nearby curve in the same homotopy class and hence the induced connection will have the same holonomy. To obtain isomonodromic deformations, we therefore need to study the deformation theory of such curves. Proposition 3. Let p Ă Z be a rational curve meeting Y transversally at four points. Then P belongs to a smooth four-parameter family of rational curves on which the cross-ratio of the points is nonconstant function. Proof. Th proof is standard Kodaira-Spencer deformation theory. By hypothesis P meets the anticanonical divisor Y in four points, so the degree of KZ on P is -4. Hence, in N is the normal bundle of P – CP1 , deg N “ ´ deg KZ ` deg KP “ 2 and so N – Opmq ‘ Op2 ´ mq for some integer m. However, since C is transversal to the 2-dimensional orbit Y of S Lp2, Cq, the map α always maps onto the normal bundle to C. We therefore have a surjective homomorphism of holomorphic vector bundles β:ObgÑ N and this implies that 0 ď m ď 2. As a consequence, H 1 pP, Nq “ 0 and H 0 pP, Nq is four-dimensional, so the existence of a smooth family follows fro Kodaira [9]. 161 Since β is surjective, its kernel is a line bundle of degree-deg N “ ´2, so we have an exact sequence of sheaves: Op´2q Ñ O b g Ñ N. Under α, the kernel maps isomorphically to the sheaf of sections of the tragent bundle T P which vanish at the four points P X Y. From the long exact cohomology sequence we have δ

0 Ñ g Ñ H 0 pP, Nq Ý Ñ H 1 pP, Op´2qq Ñ 0 188

Poncelet Polygons and the Painlev´e Equations

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and since H 0 pP, Nq is 4-dimensional and g is 3-dimensional, the map δ id surjective. But αδ is the Kodarira-Spencer map for deformations of the four points on P, so since it is non-trivial, the cross-ratio is nonconstant. Example. As the reader may realize, the situation here is very similar to the study of twistor spaces and twistor lines, and indeed there is a differential geometric context for this (see [6], [7]). This is not the agenda for this paper, but it is a useful example to see the standard twistor space-CP3 and the straight linex in it-within the current context. Let V be the 4-dimensional space of cubic polynomials ppzq “ c0 ` c1 z ` c2 z2 ` c3 z3 and consider V as a representation space of S Lp2, Cq under the action ˙ ˆ az ` b pcz ` dq3 . ppzq ÞÑ p cz ` d This is the unique (up to isomorphism) 4-dimensional irreducible representation of S Lp2, Cq. Then Z “ PpVq “ C¶3 is a compact threefold with an action of S Lp2, Cq and moreover the open dense set of cubics with distinct roots in an orbit. This follows since given any two triplex of distinct ordered points in CP1 , there is a unique element of PS Lp2, Cq which takes one to the other. However, the cubic polynomial determines an unordered triple of roots, and hence the stabilizer in PS Lp2, Cq is the symmetric group S 3 . Thinking of this as the symmetries of an equilateral triangle, the holonomy group Γ Ă S Lp2, Cq of the connection A “ α´1 is the binary dihedral group D˜ 3 . The lowerdimensional orbits consist firstly of the cubics with one repeated root, which is 2-dimensional, and those with a triple root, which constitute 162 a rational normal curve in CP3 . Together they form the discriminant divisor Y, the anticanonical divisor discussed above. 189

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A generic line in CP3 , generated by polynomials ppzq, qpzq meets Y at those values of t for which the discriminant of tppzq ` qpzq vanishes, i.e. where tppzq ` qpzq “ 0

tp1 pzq ` q1 pzq “ 0 have a common root. This occurs for t “ ´qpαq{ppαq where α is a root of the quartic equation p1 pzqapzq ´ ppzqq1 pzq “ 0 and so the line meets Y in four generically distinct points. Thus the 4-parameters family of lines in CP3 furnish an example of the above proposition. As we remarked above, this is an example of an isomonodromoc deformation, as would be any family of curves P in Proposition 3. It yields a solution of the Painlev´e equation either by applying the argument of Theorem 1 to the connection with logarithmic singularities on Z, or appealing to the universality of Malgrange’s construction. We shall not derive the solution of the Painlev´e equation here from CP3 , since it will appear via a different compactification in the context of Poncelet polygons. There we shall also see how a striaght line in CP3 defines a pair on conics with the Poncelet property for triangles.

4 Poncelet polygons and projective bundles In this section we shall study a particular class of equivariant compactifications, originally due to Schwarzenberger [16]. Consider the complex surface CP1 ˆ CP1 and the holomorphic involution σ which interchanges the two factors. The quotient space is CP2 . A profitableway of viewing this is a the map which assigns to a pair of complex numbers the coefficients of the quadratic polynomial which has them as roots. In affine coordinates we have the quotient map π : CP1 ˆ CP1 Ñ CP1 190

Poncelet Polygons and the Painlev´e Equations

191

pw, zq ÞÑ p´pw ` zq, wzq. From this it is clear that π is a double covring branched over the image 163 of the diagonal, which is the conic b Ă CP2 with equation 4y “ x2 . Moreover the line tau ˆ CP1 Ă CP1 ˆ CP1 maps to a line in CP2 which meets B at the single point πpa, aq. The images of the two lines tau ˆ CP1 and CP1 ˆ tbu are therefore the two tangents to the conic B from the point πpa, bq. Now let Opk, lq denote the unique holomorphic line bundle of bidegree pk, lq on CP1 ˆ CP1 , and define the direct image sheaf π˚ Opk, 0q on CP2 . This is a locally free sheaf, a rank 2 vector bundle Vk , and we may form the projective bundle PpVk q, a complex 3-manifold which fibres over CP3 p : PpVk q Ñ CP2

with fibres CP1 . Clearly the diagonal action of S Lp2, Cq on CP1 ˆ CP1 induces an action on PpVk q. Take a point z P PpVk q and consider its stabilizer. If ppzq P CP2 zB, then ppzq “ πpa, bq where a ‰ b. Consider the projective bundle pulled back to CP1 ˆ CP1 . The point pa, bq is off the diagonal in CP1 ˆ CP1 . so the fibre of ppVk q “ Ppπ˚ Opk, 0qq is PpOpk, 0qa ‘ Opk, 0qb q.

(7)

The stabilizer of pa, bq in S Lp2, Cq is on 3-dimensional, and acts on pu, vq P Opk, 0qa ‘ Opk, 0qb as pu, vq ÞÑ pλk u, λ´k vq. Thus, as long as u ‰ 0 or v ‰ 0, the stabilizer of the point represented by pu, vq in the fibre in finite. Thus the generic orbit is three-dimensional. We have implicitly just defined the divisor Y of lower-dimensional orbits, but to be more prescise, we have the inverse image of the branch locus D1 “ π´1 pBq as one component. The other arises from the direct image construction as follows. 191

192

N. J. Hitchin Recall that by definition of the direct image, for any open set U Ď

CP2 ,

H 0 pU, Vk q – H 0 pπ´1 pUq, Opk, 0qq so that there is an evaluation map ev : H 0 pπ´1 pUq, π˚ vk q Ñ H 0 pπ´1 pUq, Opk, 0qq. The kernel of this defines a distinguished line sub-bundle of π˚ pVk q and thus a section of the pulled back projective bundlePpVk q. This copy oc CP1 ˆ CP1 in PpVk q is a divisor D2 . 164 Both divisors are components of the anticanonical divisor Y, and it remains to check the multiplicity. Now let U be the divisor class of the tautological line bundle over the projective bundle PpVk q..The divisor D2 is a section of PpVk q pulled back to CP1 ˆ CP1 , and from its definition it is in the divisor class p˚ p´Uq ` Opk, 0q. Thus in PpVk q, D2 „ ´2U ` kH

(8)

where H is the divisor class of the pull-back by π of the hyperplane bundle on CP2 . Clearly, since B is a conic, D1 „ 2H.

(9)

Now from Grothendieck-Riemann-Roch applied to the projection π, we find c1 pVk q “ pk ´ 1qH, from which it is easy to see that the canonical divisor class is K „ 2U ´ pk ` 2qH so since ´k „ ´2U ` pk ` 2qH „ D1 ` D2 , the multiplicity in 1 for each divisor and we can take Z “ PpVk q as an example of an equivariant compactification to which Proposition 2 applies. The stabilizer of a point in ZzY is in this case the binary dihedral group D˜k , which is the inverse image in S Up2q of the group of symmetries in S Op3q – S Up2q{ ˘ 1 of a regular plane polygon with k sides. Although this can be seen quite easily from the above description of the action, there is a direct way of viewing ZzY ´ PpVk qzD1 Y D2 as the S Lp2, Cq orbit of a plane polygon. 192

Poncelet Polygons and the Painlev´e Equations

193

Note that a polygon centred on 0 P C3 is described by a non-null axis orthogonal to the plane of the polygon, and by k (if k is odd) or k{2 (if k is even) equally spaced axes through the origin in that plane. Now, given a point z P PpVk qzD1 Y D2 , its projection ppzq “ x P PpC3 qzB is a non-null direction in C3 which we take to be the axis. To find the other axes we use two facts: • The map s ÞÑ sk from Op1, 0q to Opk, 0q defines a rational map mk : PpV1 q Ñ PpVk q of degree k. • The projective bundle PpV2 q is the projectivized tangent bundle PpT q of CP2 . The first fact is a direct consequence of the definition of the direct image sheaf: H 0 pU, Vk q – H 0 pπ´1 pUq, Opk, 0qq

for any open set U Ď CP2 . The second can be found in [16]. Given these two facts, consider the set of points m2 pm´1 k pzqq Ă PpV2 q.

Depending on the parity of k this consists of k or k{2 points in PpT q all of which project to x P CP2 . In other words they are line through x or, using the polarity with respect to the conic B. points on the polar line of x. Reverting to linear algebra, these are axes in the plane orthogonal to x. We now need to apply Proposition 3 to this particular set of examples to find rational curves which meet the divisor Y “ D1 ` D2 transversally in four points. Now if P is such a curve, then the intersection number P. D1 ď 4 so ppPq “ C is a plane curve of degree d which meets the branch conic B in 2d ď 4 points, hence d “ 1 or 2. We consider the case d “ 2 first. The curve C is a coniv in CP2 . The set of all conics forms a 5-parameter family and we want to determine the 4-parameter family of conics which lift to PpVk q. Theorem 2. A conic C Ă CP2 meeting B transversally lifts to PpVk q if and only if there exists a k-sided polygon inscribed in C and circumscribed about B. 193

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Proof. A lifting of C is a section PpVk q over C, or equivalently a line subbundle M Ă Vk over C. Since C is a conic, the hyperplane bundle H is of degree 2 on C, so we can write M – H n{2 for some integer n. The inclusion M Ă Vk thus defines a holomorphic section s of the vector bundle Vk b H ´n{2 over C. But Vk is the direct image sheaf of Opk, 0q, so we have an isomorphism ˜ Opk, 0q b π˚ pH ´n{2 qq H 0 pC, Vk b H ´n{2 q – H 0 pC, where C˜ “ π´1 pCq Ă CP1 ˆ CP1 is the double covering of the conic C branched over its points of intersection with B. But it is easy to see that π˚ pHq – Op1, 1q so on C˜ we have a holomorphic section s˜ of Opk ´ n{2, ´n{2q. We have more, though, for since the intesection number´K.P “ pD1 ` D2 q.P “ 4 and D1 .P “ B.C “ 4, P lies in PpVk qzD2 , where D2 was given as the kernel of the evaluation map. It the section s˜ vanishes anywhere, then the section of PpVk q will certainly meet D2 , thus s˜ is everywhere non-vanishing an Opk ´ n{2, ´n{2q is the trivial bundle. In ˜ Now C is a conic, so C˜ is the divisor particular, its degree is zero on C. ˚ 2 166 of a section of π pH q – Op2, 2q and so the degree of the line bundle is 2k ´ 2n “ 0 and thus n ´ k. Hence a conic in CP2 lifts to Ppvk q if and only if it has the property that ˜ Opk{2, ´k{2q – O on C. Now recall the Poncelet problem [3]: to find a polygon with k sides which is inscribed in a conic C and circumscribed about a conic B. The projection π : CP1 ˆ CP1 Ñ CP2 we have already used is the correct setting for the problem. Let pa, bq be a point in CP1 ˆ CP1 and consider the two lines tau ˆ 1 CP and CP1 ˆ tbu passing through it. The first line is a divisor of the linear system Op1, 0q and the second of Op0, 1q. As we have seen, their images in CP2 are the two tangents to the branch conic B from the point πpa, bq. Now let C be the conic which contains the vertices of the 194

Poncelet Polygons and the Painlev´e Equations

195

Poncelet polygon, and let P1 “ pa1 , b1 q P C˜ Ă CP1 ˆ CP1 be a point lying over an initial vertex. The line ta1 u ˆ CP1 meets C˜ „ Op2, 2q in two points generically,. which are P1 and a second point P2 “ pa1 , b2 q. The two points πpP1 q and πpP2 q lie on C, and the line joining them is pipta1 uˆCP1 q which is tangent to B, and hence is a side of the polygon. The other side of the polygon through pipP2 q is πpCP1 ˆ tb2 uq which meets the conic C at πpP3 q “ πpa2 , b2 q. We carry on this procedure using the two lines through each point, to obtain P1 , . . . , Pk`1 . Since the Poncelet polygon is closed with k vertices, we have πpPk`1 q “ πpP1 q. Consider now the divisor classes Pi ` Pi`1 . We have P1 ` P2 „ Op1, 0q

P2 ` P3 „ Op0, 1q

P3 ` P4 „ Op1, 0q ...

...

and Pk ` Pk`1 „“ calO1, 0 if k is odd and „ Op0, 1q if k is even. In the odd situation, taking the alternating sum we obtain P1 ` Pk`1 „ Oppk ` 1q{2, ´pk ´ 1q{2q

(10)

and since πpPk`1 “ πpP1 qq, then Pk`1 “ P1 or σpP1 q. However, in the former case, we would have Pk ` P1 „ Pk`1 „ Op1, 0q „ P1 ` P2 and consequently P2 „ Pk on the elliptic curve C˜ which implies P2 “ 167 Pk . But πpPk q and πpP2 q and πpP2 q are different vertices of the polygon, so we must have Pk`1 “ σP1 . This that the divisor Pk`1 ` P1 “ π´1 pπpP1 qq ans so in the notation above Pk`1 ` P1 „ H 1{2 “ Op1{2, 1{2q. From (10) we therefore obtain the constraint on C˜ Opk{2, ´k{2q „ O 195

(11)

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which is exactly the condition for the conic to lift to PpVk q. A similar argument leads to the same condition for k even, where in this case Pk`1 “ P1 . In the case that d “ 1, C is a line, but the argument in very similar. Here M – H n for some n and on C˜ we have a section ξ of Opk ´ n, ´nq. This time, since P.D2 “ 2, the line bundle is of degree 2, so k ´ 2n “ 2, ˜ and so a lifting is defined by a section of O1 ` k{2, 1 ´ k{2 on C. Example. Let us now compare this interpretation with the equivariant compactification CP3 of S Lp2, Cq{D˜ 3 discussed earlier. In the first place, consider the line bundle U˜ “ U ´ 2H on PpV3 q.Now since for any 2-dimensional vector space V ˚ – V b ^2 V ˚ , PpVk q “ PpVk˚ q, but with different tautological bundles. The ˜ and so there are canonical tautiological bundle for PpV3˚ q is actually U, isomorphisms ˜ – H 0 pCP2 , V3 q – pCP1 ˆ CP1 , Op3, 0qq H 0 pPpV3 q, ´Uq – H 0 pCP1 , Op3qq – C4 .

˜ therefore maps PpV3 q equivariantlu to CP3 . The linear system | ´ U| Since P.D1 “ 4 and P.D2 “ 0, it follows from (8) and (9), that P.H “ 2 and P.U “ 3, and so P.U˜ “ ´1, so under this mapping the curves P map to projective lines.

168

There is a more geometic way of seeing the relation of lines in CP3 to Poncelet triangles. Recall that we are viewing CP3 as the space of cubic polynomials, and CP2 as the space of quadratic polynomials. The quadraitcs with a fixed linear factor z´α describe, as we have seen, aline in CP2 which is tangent to the discriminant conic at the quadratic pz ´ αq2 . Thus the three linear factors of a cubic pz ´ αq,pz ´ βq,pz ´ γq, pz ´ γqpz ´ αq. 196

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Poncelet Polygons and the Painlev´e Equations

Now consider a straight line of cubics pt pzq “ tppzq ` qpzq with roots αt, betaptq and γt. We have a 1-parameter family of triangles and tppαq ` qpβq “ 0

tppβq ` qpβq “ 0.

Now from these two equations 0 “ ppαqqpβq ´ ppβqqpαq “ pα ´ βqrpα, betaq where rpα, βq is a symmetric polynomial in α, β. It is in fact quadratic in αβ, αβ and thus defines a conic C in the plane. Hence, as t varies, the vertices of the triangle lie on fixed conic C, and we have a solution of the Poncelet problem for k “ 3.

5 Solutions of Painlev´e VI To find more about the connection we have just defined on Z “ PpVk q entails descending to local coordinates, which we do next. Consider the projective bundle PpVk q pulled back to CP1 ˆ CP1 . At a point off the diagonal pa, bq P CP1 ˆ CP1 , as in (7), the fibre is PpOpk, 0qa ‘ Opk, 0qb q “ PpOpk, 0q ‘ Op0, kqqa,b and awat from the zero section of the second factor, this is isomorphic to Opk, ´kqa,b .

Now choose standard affine coordinates pw, zq in CP1 ˆ CP1 . Since KCP1 – Op´2q, we have corresponding local trivializations dw and dz of Op´2, 0q and Op0, ´2q. These define a local trivialization pdwq´k{2 pdzqk{2 of calOk, ´k, and thus coordinates k{2

pw, z, sq ÞÑ spdwq´k{2 pdzqpw,zq .

197

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Note that Z is the quotient of this space by the involution pw, z, sq ÞÑ pz, w, s´1 q. From this trivialization, the natural action of S Lp2, Cq on differentials gives the action on Z: ˆ ˙ pcz ` dqk aw ` b az ` b pw, z, sq ÞÑ s . , , cw ` d cz ` d pcw ` dqk Differenting this expression at the identity gives the tangent vector pw1 , z1 , s1 q corresponding to a matrix ˆ 1 ˙ a b1 Pg c1 ´a1

as

w1 “ ´c1 w2 ` 2a1 w ` b1 z1 “ ´c1 z2 ` 2a1 z ` b1

s1 “ ´kc1 pw ´ zqs

This is αpa1 , b1 , c1 q P T Zpw,z,sq . Solving for pa1 , b1 , c1 q gives the entries of the matrix of 1-forms A “ α´1 as pw ` zqds dw ´ dz ´ 2pw ´ zq 2kspw ´ zq wzds wdz ´ zdw ` “ pw ´ zq kspw ´ zq ds “´ kspw ´ zq

A11 “ A12 A21

(12)

Proposition 4. The resedue of the connection at a singular point is conjugate to ˙ ˙ ˆ ˆ 1{2k 0 1{4 0 on D2 on D1 and 0 ´1{2k 0 ´1{4 Proof. In these coordinates, s “ 0 is the equation of D2 . From (2), the residue of A at s “ 0 is ˆ ˙ ´pw ` zq{2kpw ´ zq wz{kpw ´ zq (13) ´1{kpw ´ zq pw ` zq{2kpw ´ zq 198

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199

which has determinant ´1{4k2 and therefore eigenvalues ˘1{2k. To find the rediduce at D1 , we need different coordinates, since the above ones are invalid on the diagonal. Take the affine coordinates x “ ´pw ` zq, y “ wz on CP2 . Since the holomorphic functions in w, z form 170 a module over the symmetrix functions generated by 1, w ´ z we can use these to give coordinates in the projectivized direct image PpVk q, which are valid for w “ z. We obtain an affine fibre coordinate t related to s above by t`w´z s“ . t´w`z Using this ans local coordinates x and u “ pw ´ zq2 “ x2 ´ 4y on CP2 the divisor D1 is given by u “ 0 and the residue here is ˆ ˙ 1{4 ` x{2kt x{4 ` x2 {4kt (14) ´1{kt ´1{4 ´ x{2kt This has determinant ´1{16 and hence eigenvalues ˘1{4.

Remark . Exponentiating the residues we see that the holonomy of a small loop around the divisor D1 or D2 is conjugate to: ˙ ˆ ˆ iπ{k ˙ i 0 e 0 on D2 on D1 0 ´i 0 e´iπ{k In the dihedral group Dk Ă S Op3q the conjugacy classes are those of a reflection in the plane and a rotation by 2π{k. These facts tell us something of the structure of the divisor D1 . Since we know that the residue of the meromorphic connection at a singular point lies in the Lie algebra of the stabilizer of the point, and this is here semisimple, the orbit is isomorphic to S Lp2, Cq{C˚ – CP1 ˆ CP1 z∆ where ∆ is the diagonal. The projections onto the two factors must, by S Lp2, Cq invariance, be the two eigenspaces of the residue corresponding to the eigenvalues ˘1{4. 199

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Now D1 “ π´1 pBq is a projective bundle over the conic B – CP1 . By invariance it must be one of the factors above. To see which, note that from (14), eigenvectors for the eigenvalues 1/4 and -1/4 are respectively. ˆ ˙ ˆ ˙ x ` kt x and ´2 ´2 and so, from the choice of coordinates above, clearly the second represents the projection ot B. Note, moreover, that on the diagonal w “ z, 171 the coordinate x “ ´pw ` zq “ ´2z, so that x is an affine parameter ˆ ˙ 1 1 on B – ∆ – CP . Furthermore, when x “ 8 the vector is an 0 eigenvector of the residue with eigenvalue -1/4. Now let us use this information to determine the solution to tyhe Painlev´e equation corresponding to a rational curve P Ă Z. Recall that the curve C “ πpPq is a plane curve of degree d, where d “ 1 or d “ 2. As we have seen,when d “ 1, any line is of this form, but when d “ 2, the conic must circumscribe a Poncelet polygon. By the S Lp2, Cq action, we can assume that C meets the conic B at the poitn x “ ´8. From the discussion above, if A8 is the residue of the connection at this point, then ˆ ˙ ˆ ˙ 1 1 1 A8 “´ 0 4 0 From Proposition 1, the solution of the Painlev´e equation is the point y on the curve P at which Apyq has this same eigenvector, i.e. where A21 pyq “ 0 Proposition 5. A line in the plane defines a solution to Painlev´e sixth equation with coefficients pα, β, γ, δq “ p1{8, ´1{8, 1{2k2 , 1{2´1{2k2 q. A Poncelet conic in the plane defines a solution to the Painlev´e equation with coefficients pα, β, γ, δq “ p1{8, ´1{8, 1{8, 3{8q. Proof. The residuces on D1 and D2 are given by (14) and (13). The lifting of a line meets D1 and D2 in two points each, so using (6) (and 200

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201

taking account of the fact that the roles of the two basis vectors ar interchanged), we obtain the first set of coefficients. The lifting of a Poncelet conic meets D1 in four points, which gives the second set, agin from (6). Proposition 6. The lifting of a line in CP3 to PpVk q defines the solution ? y“ x of Painlev´e VI with pα, β, γ, δq “ p1{8, ´1{8, 1{2k2 , 1{2 ´ 1{2k2 q. Proof. Taking tha double covering C˜ Ă CP1 ˆCP1 of the line C and the 172 coordinates w, z, s on the corresponding covering of Z, the lifted curve ˜ in fact , as we shall P id defined locally by a function s on the curve C. see next, s is a meromorphic function on C˜ with certain properties. From the comments following Theorem 2, the lifting is given by a holomorphic section ξ of Op1 ` k{2, 1 ´ k{2q. This line bundle has ˜ and so ξ vanishes at two points. Applying the involution degree 2 on C, ˚ σ, the σ ξ is a section of Op1 ´ k{2, 1 ` k{2q. Considering ξ as a section of Opk, 0q b Op1 ´ k{2, 1 ` k{2q. Considerign ξ as a section of Opk, 0q b Op1 ´ k{2, 1 ´ k{2q, teh lifting of C to PpVk q is defined by pξa,b , ξb,a q, or in the coordinates w, z, s, spdwq´k{2 pdzqk{2 “ ξ{σ˚ ξ.

(15)

Since dw´1{2 and dz´1{2 are holomorphic sections of Op1, 0q and Op0, 1q, ˜ s is a meromophic function. Now using the S Lp2, Cq it follows that on C, action, we may assume that the line C is given by x “ 0, which means that C˜ has equation w “ ´z which defines as obvious trivialization of Opk, ´kq and from which we deduce that s has two simple zeros at pa1,´a1 q, pa2 , ´a2 q and two poles ˜ wer obtain, at p´a1 , a1 q, p´a2 , a2 q. Using z as an affine parameter on C, up to a constant multiple, s“

pz ` a1 qpz ` a2 q . pz ´ a1 qpz ´ a2 q 201

(16)

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N. J. Hitchin

Now y “ wz is an affine parameter on the line C, which meets the conic B at y “ 0, 8. The lifting P meets the divisor D2 where y “ ´a21 and ? 2 y “ ´a2 , so putting a1 “ i and a2 “ ´x then Pm is a projective line with a parametrization such that the singular points of the induced connection are 0, 1, x, 8, as required for the Painlev´e equation. It ramians to determine the solution of the equation, which is given by A21 pyq “ 0. But from (12), this is where ds “ 0, and from (16) this is equivalent to 1 1 1 1 ´ ` ´ “0 z ` a1 z ´ a1 z ` a2 z ´ a2 which gives y “ ´z2 “ ´a1 a2 “

?

x

with the above choices of a1 , a2 .

Remarks. 173

? 1. By direct calculation, the fucntion y “ x solves Painlev´e VI for any coefficient satisfying α ` β “ 0 and γ ` δ “ 1{2. From (6) this occurs when the residues are conjugate in pairs. 2. When k “ 2, we obtain pα, β, γ, δq “ p1{8, ´1{8, 1{8, 3{8q which are the coefficients arising from Poncelet conics. We shall see the same solution appearing in the next section in the context of Poncelet quadrilaterals. Naturally, the solutions corresponding to Poncelet conics are more complicated, and we shall give some explicitly in Section 6. Here we give the general algebraic procedure for obtaining them. In the case of a conic C in CP2 , we have a section ξ, in fact a trivial˜ As in (15), ization, of the bundle Opk{2, ´k{2q on the elliptic curve C. we still define the lifting by spdwq´k{2 pdzqk{2 “ ξσ˚ ξ but in this case ξ is non-vanishing. The section pdwq´k{2 vanishes to order k at 8 P CP1 , and so at the two points p8, in f tyq, p8, bq where C˜ 202

Poncelet Polygons and the Painlev´e Equations

203

meets t8u ˆ CP1 . Similarly pdzq´k{2 vanishes at p8, 8q, pb, 8q. The meromorphic function s can be regarded as a map oc curves s : C˜ Ñ CP1 . It follows that s is a meromorphic function on C˜ with a zero of order k at pb, 8q, a pole of order k at p8, bq and no other zeros or poles. ´1 The derivative ds in invariantly defined as a section of Kc˜ bs˚ KCP ´1 – ˚ ˜ s Op2q (since C is an elliptic curve and hence has trivial canonical bundle). In particular ds vanishes with total multiplicity 2k. But since p8, aq and pa, 8q are branch points of order k, ds has a zero of order k ´ 1 at each of these points, leaving two extra points as the remaining zeros. Since the involution σ takes s to s´1 , these points are paired by the involution, and give a single point y P CP1 which is our solution to the Painlev´e equation. Fortunately Cayley’s solution in 1853 to the Poncelet problem gives us means to find y algebraically. A usefull modern account of this is given by Griffiths and Harris in [5], but the following description I owe to M.F. Atiyah. Suppose the elliptic curve C˜ is described as a cubic in CP2 given by 2 v “ hpuq, where hpuq is a cubic in polynomial and hp0q “ c20 ‰ 0. We shall find the condition on the coefficients of h in order that there should exist a polynomial gpv, uq of degree pn ´ 1q (a section of Opn ´ 1qq on the curve with a zero of order p2n ´ 1q at pv, uq “ pc0 , 0q and a pole of order pn ´ 2q at u “ 8. Given such a polynomial, 174 spv, uq “

gpv, uq gp´v, uq

is the function on the curve (for k “ p2n ´ 1q) used above, and the zeros of its derivative define the solution ypxq to the Painlev´e equation. A very similar procedure dealsawith the case of even k. To find g, expand hpuq as a power series in z, making a choice c0 of square root of hp0q: v “ c0 ` c1 u ` c2 u2 ` ¨ ¨ ¨ 203

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and the put vm “ c0 ` c1 u ` ¨ ¨ ¨ ` cm´1 um´1 . Now clearly v ´ vn has a zero of order n on the curve at u “ 0, as do other functions constructed from the vm : v ´ vn “ cn un ` ¨ ¨ ¨ ` c2n´2 u2n´2 ` ¨ ¨ ¨

upv ´ vn´1 q “ cn´1 un ` ¨ ¨ ¨ ` c2n´3 u2n´2 ` ¨ ¨ ¨

u2 pv ´ vn´2 q “ cn´2 un ` ¨ ¨ ¨ ` ¨ ¨ ¨ ...

n´2

u

“ ...

pv ´ v2 q “ c2 un ` ¨ ¨ ¨ ` cn u2n´2 ` ¨ ¨ ¨

We can then find n ´ 1 coefficients λ0 , λ1 , . . . , λn´2 such that gpv, uq ” λ0 pv ´ vn q ` λ1 upv ´ vn´1 q ` ¨ ¨ ¨ ` λn´2 un´2 pv ´ v2 q vanishes at u “ 0 to order 2n ´ 1 if and only if » cn cn´1 — cn`1 cn det M “ 0 where M “ — – ... ... c2n´2 c2n´3

ﬁ . . . c2 . . . c3 ﬃ ﬃ . . . . . .ﬂ . . . cn

(17)

This is Cayley’s form of the Poncelet constraint. If (17) holods, gpv, uq is a polynomial of degree n ´ 1 which, upon inspection, vanishes with multiplicity n ´ 2 at the inflexion point at infinity of the curve. Its total intersection number with the cubic C˜ is 3pn ´ 1q “ p2n ´ 1q ` pn ´ 2q, so there are no more zeros. Thus the condition det M “ 0 is necessary and sufficient for the construction of the required function s with a zero of order k at pv, uq “ pc0 , 0q and a pole of order k at pv, uq “ p´c0 , 0q. 175 In the case of a pair of conics in the plane, defined by symmetric matrices B and C,. the constraint is on the cubic hpuq “ detpB ` uCq in order for the conics to satisfy th Poncelet condition. Note that gpv, uq “ ppuqv ` qpuq 204

Poncelet Polygons and the Painlev´e Equations

205

where p and q polynomials of degree n ´ 2 and n ´ 1 respectively. Thus s“

ppuqv ` qpuq ´ppuqv ` qpuq

and ds vanishes if pqv1 ` pp1 q ´ pq1 qv “ 0.

Using v2 “ hpuq, this is equivalent to

rpuq ” ppuqqpuqh1 puq ` 2pp1 puqqpuq ´ ppuqq1 puqqhpuq “ 0 This is a polynomial in u of degree n2n ´ 1, which by construction vanishes to order k ´ 1 ´ 2n ´ 2 at u “ 0. It is thus of the form rpuq “ au2n´2 pu ´ bq and so y, the solution to the Painlev´e equation which corresponds to a zero of ds, is defined in terms of the ration of the two highest coefficients of rpuq. Since the solution the solution to the Painlev´e equation has singularities at the four points 0, 1, x, 8, a M¨obius transformation gives the variable x, and the solution ypxq, as: x“

e3 ´ e1 e2 ´ e1

y“

b ´ e1 e2 ´ e1

(18)

where e1 , e2 and e3 are the roots of hpuq “ 0. To calculate b expalicitly is easy. Putting ppuq “ p0 ` P1 u ` ¨ ¨ ¨ ` Pn´2 un´2 and qpuq “ qo ` q1 u ` ¨ ¨ ¨ qn´1 un´1 and looking at the coefficients of rpuq, we find qn´2 pn´3 ´3 pn´2 qn´1 λn´3 λ0 ` λ1 cn´3 ` ¨ ¨ ¨ ` λn´2 c0 “ ´3 λn´2 λ0 cn´1 ` λ1 cn´2 ` ¨ ¨ ¨ ` λn´2 c1

b“

from the definition of ppuq and qpuq. Now the coefficient λi are the entries of column vector λ such that Mλ “ 0. Thus in the generic case where the rank of M is n ´ 3, these are given by cofactors of M. We can 176 205

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N. J. Hitchin

then write cn cn`1 . . . c2n´3 b“´ cn cn`1 . . . c2n´3

. . . c4 c2 cn´1 cn . . . c5 c3 ... . . . c2n´3 c2n´4 ... ... . . . . . . cn`1 cn´1 cn´2 cn´3 ´ 3 cn´1 . . . c3 cn´1 cn . . . cn . . . c4 ... . . . . . . . . . c2n´3 c2n´4 cn´1 cn´2 c2n´4 . . . cn

. . . c2 . . . . . . . . . cn´1 . . . c0 (19) . . . c2 . . . . . . . . . cn´1 . . . c1

This effectively gives us a concrete form for the solution of the Painlev´e equaton for k pdd, through we shall try to be more explicit in special cases in the nest section. When k is even, a similar analysis can be applied. Very briefly, if k “ 2n and n ď 2, then the vanishing of cn Cn`1 cn`2 cn`1 ... . . . c2n´1 . . .

. . . c3 . . . c4 . . . . . . . . . cn`1

is the condition for the existence of λi so that gpv, uq “ λ0 pv ´ vn`1 q ` λ1 upv ´ vn q ` ¨ ¨ ¨ ` λn´2 u3 pv ´ v3 q has a zero of order 2n at u “ 0. The rest follows in a similar manner to the above.

6 Explicit solutions We shall now calculate explicit solutions to Painlev´e VI with coefficients pα, β, γ, δq “ p1{8, ´1{8, 1{8, 3{8q for small values of k. Clearly, from the interpretation in terms of Poncelet polygons, we must have k ď 3. The discussion in the previous section shows that we need to perform calculations with the coefficients of the cubic hpuq “ detpb ` uCq where 206

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Poncelet Polygons and the Painlev´e Equations

B and C are symmetric matrices representing the conics we have denoted with the same symbol. For convenience, we take the cubic hpuq “ p1 ` px1 ` x2 quqp1 ` px2 ` x0 quqp1 ` px0 ` x1 quq “ 1 ` 2s1 u ` ps21 ` s2 qu2 ` ps1 s2 ´ s3 qu3

(20)

where si is the ith elementary symmetric function in x0 , x1 , x2 . 6.1 Solutions for k “ 3

177

For k “ 3 the Poncelet constraint from (17) is c2 “ 0, which is s2 “ 0 for the above cubic, and can therefore be writtne 1 1 1 ` ` “ 0. x0 x1 x2 This is the equation in homogeneous coordinates px0 , x1 , x2 q for a plane conic which can clearly be parametrized rationally by x0 “

1 1`s

x1 “

´1 s

x2 “ ´1

(21)

The polynomial g is just gpv, uq “ v ´ p1 ` s1 uq, and this gives rpuq “ s1 s3 u3 ` 3s3 u2 and hence b “ ´3{s1 . Substituting the parametrization (21) in (18) and using the fact that the roots hpuq “ 0 are u “ ´1{px1 ` x2 q etc, gives the solution ypxq to the Painlev´e equation ad y“

s2 ps ` 2q ps2 ` s ` 1q

where

6.2 Solutions for k “ 4

x“

s3 ps ` 2q 2s ` 1

The Poncelet constraing here is c3 “ 0, which in the formalism above is s3 “ 0, that is x0 “ 0 x1 “ 0 x2 “ 0. 207

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In CP3 this consists of three lines. Take the component x0 “ 0 and parametrize it by x1 “ 1 x2 “ s.

Now the polynomial g is given by gpv, uq “ v ´ p1 ` s1 u ` 12 s2 u2 q, which yields 1 rpuq “ s1 s22 u4 ` s22 u3 2 and hence b “ ´2{s1 . Substituting the parametrization, we obtain x “ s2 , y “ s, thus the solution to the Painlev´e equation is ? ypxq “ x. ? 178 Remark. In Proposition 6, we saw that the same solution y “ x with coefficients pα, β, γ, δq “ p1{8, ´1{8, 1{8, 3{8q arises from taking an curve P in PpV2 q with P.D1 “ P.D2 “ 2, and hence has holonomy in D2 , which is the quaternion group t˘1, ˘i, ˘ j, ˘ku, a proper subgroup of D˜ 4 . Recall also that from [16], PpV2 q – PpT q, the projectivized tangent bundle of CP2 together with a line passing through it, or equivalently a line with a distinguished point. Thus there is a projection also to the dual projective plane CP2˚ . In other words PpT q “ tpp, Lq P CP2 ˆ CP2˚ : p P Lu with projections onto the two factors. In the terminology above, the two corresponding hyperplane divisor classes are H and H ´ U. Now the curve P which defines the solution to the Painlev´e equation satisfies P.H “ P.pH ´ Uq “ 1. It follows easily that P is obtained by taking a point q P CP2 and a skew line M. The set P “ tpp, Lq P PpT q : q P L and p “ L X Mu describes the rational curve P Ă PpT q. According to our formula, this curve must correspond to a Poncelet conic. In act, letpp, Lq be a point of P, let ℓ denote the pole of the line L with respect to the conic B. The line ℓp has pole q, and as L varies in the pencil of lines through p, the point q describes a conic C. If L 208

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is a tangent through p to the conic B, then q4 is the point of contact B X L. Let q1 and q2 be the two points of contact of the tangents through p, then the conic C passes through p, q1 and q2 . But pq1 pq2 is then a degenerate Poncelet quadrilateral, and by Poncelet’s theorem [3], if there is a Poncelet polygon through one point of C, then there exists one through each point. 6.3 Solution for k “ 5 From (17), the Poncelet constraint is „ c3 c2 det “0 c4 c3 which in terms of the symmetric functions si is 4s23 ` s22 ´ 4s1 s2 s3 “ 0

(22)

Now from (19), b“´

pc3 ´ c1 c2 q c2 ´3 c3 pc1 c3 ´ c22 q

“

ps3 ` s1 s2 q s2 ´6 s3 p2s1 s2 ` s22 q

and using (22) this becomes

179

b “ ´20

s2 s3 ` 3s32 q

p4s23

It is convenient to introduce coordinates u, v by setting x0 “ 1 u “

1 1 ` x1 x2

v“

1 x1 x2

and then the constraint (22) becomes v“ and ´5

p1 ` uqp1 ´ uq2 4u

pu ` 1qpu ´ 1q2 p3upu ` 1q2 ` pu ´ 1q2 q 209

(23)

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N. J. Hitchin

Now 1{x1 and 1{x2 are the roots of the quadratic z2 ´ uz ` v “ 0, and so, using (23),

and now putting

a 1 1 1 , “ pu ˘ 1 ` u ´ u´1 q x1 x2 2 w2 “ 1 ` u ` u´1

(24)

we finally obtain the solution of the Painlev´e equation as ˆ ˙ pu ´ w ` 2qpu ` wq 20u2 1´ y“ pu ` w ´ 2qpu ´ wq p3upu ` 1q2 ` pu ´ 1q2 q pu ´ w ´ 2qpu ´ w ` 2qpu ` wq2 and w2 “ 1 ` u ` u´1 where x “ pu ` w ´ 2qpu ` w ` 2qpu ´ wq2 Note that (24) is the equation of an elliptic curve, so that x and y are meromorphic functions on the curve. It is a special elliptic curve, in fact under the Cremona transformation xi ÞÑ 1{xi , the equation (22) transforms into the plane cubic s31 ´ 4s1 s2 ` 4s3 “ 0 and the symmetric group S 3 clearly acts as automorphism of the curve. The study of the Poncelet constraints for smal values of k in undertaken in [1], and the reader will find that, apart from k “ 6 and k “ 8, the formulae rapidly become more complicated. We shall only consider now these two further cases. 180

6.4 Solution for k “ 6

Here, from [1], we find that the constraint factorizes px0 x1 ` x1 x2 ` x2 x0 qp´x0 x1 ` x1 x2 ` x2 x0 qpx0 x1 ´ x1 x2 ` x2 x0 q px0 x1 ` x1 x2 ´ x2 x0 q “ 0.

The first factor represent the case k “ 3 embedded in k “ 6, by thinking of a repeated Poncelet triangle as a hexagon, We choose instead the third 210

211

Poncelet Polygons and the Painlev´e Equations factor, which can be written as 1 1 1 “ ` . x0 x1 x2 This is a conic, and we rationally parametrize it by setting x0 “

1 1`s

x1 “

1 s

x2 “ 1.

After some calculation, this gives a solution to the Painlev´e equations sp1 ` s ` s2 q p2s ` 1q 3 s ps ` 2q x“ p2s ` 1q y“

where

6.5 Solution for k “ 8 Here, again referring to [1], we find the constraint equation splits into components p´x02 x12 ` x12 x22 ` x22 x02 qpx02 x12 ´ x12 x22 ` x22 x02 qpx02 x12 ` x12 x22 ´ x22 x02 q “ 0. one of which is given by the equation 1 1 1 “ 2` 2 1 x0 x1 x2 and , parametrizing this conic rationally in the usual way with x0 “ 1

x1 “

1 ` s2 2s

x2 “

1 ` s2 1 ´ s2

one may obtain the solution to the Painlev´ed equation as 4sp3s2 ´ 2s ` 1q p1 ` sqp1 ´ sq3 ps2 ` 2s ` 3q ˆ ˙4 2s x“ . 1 ´ s2 y“

where

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7 Painlev´e curves 181

The examples above show that there do indeed exist algebraic solutions to Painlev´e sixth equation, for certain values of the coefficients, despite the general description of solutions to these equations as “Painlev´e transcendants”. In general, an algebraic solution is given by a polynomial equation Rpx, yq “ 0 which defines an algebraic curve. So far, we have only seen explicit examples where this curve is rational or elliptic, but higher genus curves certainly do occur. We make the following definition: Definition 2. A Painlev´e curve is the normalization of an algebraic curve Rpx, yq “ 0 which solves Painlev´e’s sixth equation (5) for some values of the coefficients pα, β, γ, δq. Just as the elliptic curve above corresponding to the solution for k “ 5 was special, so are Painlev´e curves in general. The equation (5) was in fact found not by Painlev´e, but by R. Fuchs [4], but nevertheless falls into the Painlev´e classification by its characteristic property that its solution have no “movalble singular points”. What this means is that the branch points or essential singularities of solutions ypxq are independent of the constants of integration. In the case of Painlev´e VI, these points occur only at x “ 0, 1, 8. Now if X is a Painlev´e curve, x and y are meromorphic functions on X, and so there are no essential singularities. The function x : X Ñ CP1 is thus a map with branch points only at x “ 0, 1, 8. Such curves have remarkable properties. In the first place, it follows from Weil’s rigidity theorem[17], that X is defined over the algebraic closure Q of the rationla (from Belyi’s theorem [2] this actually characterizes curves with such functions). Secondly, by uniformizing CP1 zt0, 1, 8u, X – H{Γ 212

Poncelet Polygons and the Painlev´e Equations

213

where H is the upper half-plane and γ is asubgroup of finite index in the principal congruence subgroup γp2q Ă S Lp2, Zq. Thus, in some manner, each algebraic solution of Painlev´e VI gives rise to a problem involving elliptic curves. Coincidentally, the investigation of the Poncelet problem by Barth and Michel in [1] proceeds by studying a modular curve. This is curve which occurs in parametrizing elliptic curves with • a level-2 structure

182

• a primitive element of order k ˜ the level-2 In our model of the Poncelet problem, the elliptic curve is C, structure identifies the elements of order two (or equivalently an ordering of the branch points of π), and the Poncelet constraint (11) selects the line bundle Op1{2, ´1{2q of order k on C˜ or equivalently a point of order k on the curve, the zero of the function s in Section 5. Choosing a primitive element avoids recapturing a solution for smaller k. As described by Barth and Michel, the stabilizer of a primitive element of order k is "ˆ ˙ * a b Γ00 pkq “ P S Lp2, Zq : a ” d ” 1pkq and c ” 0pkq c d (25) Now if k is odd, matrices A P γ00 pkq can be chosen such that A mod 2 is any element of S Lp2, Z2 q. Thus γ00 pkq acts transitively on the level-2 structures. Consider in this case the modular curve X00 pk, 2q “ H{Γ00 pkq X Γp2q

(26)

Since ´I P Γ00 pkq acts trivially on H, this is a curve parametrizing opposite pairs of primitive elements of order k, and level-2 structures. When k is even, however, only the two matrices ˆ ˙ 1 0 0 1

ˆ ˙ 1 1 0 1

213

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N. J. Hitchin

are obtained by reducing mod 2 from Γ00 pkq. Thus the group has three orbits on the level-2 structures, and the curve X00 pk, 2q has three components each given by (26). There is another curve in the picture: the curve Πk defined by the Cayley constraint (17), with the ci symmetric functions of px0 , x1 , x2 q as defined by (20). This is a plane curve in homogeneous coordinates px0 , x1 , x2 q. Barth and Michel show that a birational image of X00 pk, 2q lies as a union of components of Πk . In the examples of Section 6, we have already seen this curve, connected for k “ 3, 5 but with different components for k “ 4, 6, 8. In the algebraic construction of y in Section 5, it is clear fro (19) that x and y are meromorphic functions on Πk , and so the Painlev´e curve Xk is a rational image of the modular curve. In fact, we have the following Proposition 7. The Painlev´e curve Xk defined by Poncelet polygons is birationally equivalent to the modular curve H{Γ00 pkq X Γp2q. 183

Proof. Let Yk be the modular curve,then we already have a map f : Yk Ñ Xk as described above. We shall define an inverse on the complement of a finite set. Let the Painlev´e curve be defined by the equation Rpx, yq “ 0 and suppose px, yq is a point on the curve such that BR{By ‰ 0. Then BR{Bx dy “´ dx BR{By is finite, and thus we can recover the connection matrix Apzq on CP1 zt0, 1, x, 8u, its coefficients being rational in x, y, dy{dx, by using the formulae for defining the connection from the solution of the Painlev´e equation as in [8] (cf Section 2). Now pull the connections back to the elliptic curve E which is the double covering of CP1 branched over the four points. The fundamental group of the punctured elliptice curve consists of he words of even length in the generators γ1 , γ2 , γ3 of π1 pCP1 zt0, 1, x, 8uq. But under 214

Poncelet Polygons and the Painlev´e Equations

215

the holonomy representation into S Op3q “ S Up2q{ ˘ 1 these generators map to reflections in a plane. Thus the even words map to the rotations in the dihedral group, the cyclic group Zk . Around a singular point in CP1 , the holonomy is conjugate to ˆ ˙ i 0 0 ´i and so on the double covering branched around the point, the holonomy is ˆ ˙ ´1 0 0 ´1 and hence the identity in S Op3q. Thus the holonomy is that of a smooth connection, and so defines an element of Hompπ1 pEq, Zk q. This is flat line bundle of order k, and through the constructions in Sections 4 and 5 is the same bundle as Op1{2, ´1{2q. We have actually made a choice here, since holonomy is determined up to conjugation. There is a rotation in S Op3q which takes the generating rotation of the cyclic group Zk to its inverse. Thus x, y defines a pair of k-torsion points on the elliptic curve, and hence a single point of the modular curve Yk “ X00 pk, 2q. Remark. A straightforward consideration of the branching over 0, 1, 8 leads to the formula 1 g “ pp ´ 3q2 4 for the genus g of X00 pp, 2q when p is prime (see [1]). Thus the Painlev´e 184 curve can have arbitrarily large genus.

References [1] W. Barth & J. Michel. Modular curves and Poncelet polygons, Math. Ann. 295 (1993) 25–49. [2] G. V. Belyi, On Galois extensions of a maximal cyclotomic, field, Math. USSR Izvestiya, 14 (1980) 247–256. 215

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[3] H. J. M. Bos, C. Kers, F. Oort & D. W. Raven, Poncelet,s closure theorem, Exposition. Math. 5 (1987) 289–364. ¨ [4] R. Fuchs, Uber linear homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegene wesentilich singul¨aren Stellen,Math. Ann. 63 (1907) 301–321. [5] Ph. Griffiths & J. Harris, On Cayley,s explicit solution to Poncelet,s porishm, Enseign. Math 24 (1978) 31–40. [6] N. J. Hitchin, A new family of Einstien metrics, Proceeding of conference in honour of E. Calabi, Pisa (1993), Academic Press (to appear) [7] N. J. Hitchin, Twistor spaces and isomonodromic deformations, (in preparation) [8] M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients II, Physica, 2D (1981) 407–448. [9] K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math, 75 (1962) 146–162. [10] B. Malgrange, Sur les deformations isimonodromiques I. Signgularit´es r´eguli`eres, in MNath´ematique et Physique, S´emainaire de l’Ec˙ole Normale Sup´erieure 1979-1982, Progress in Mathematics 37, Birkh¨a user, Boston (1983) 401–426. [11] V. B Mehta and C.S. Seshadri, Moduli of vector bundles on curves with parabolic structure, Math. Ann. 28 (1980) 205–239. 185

[12] S. Mukai and H. Umemura, Minimal rational 3-folds, in Algebraic Geometry (Proceedings, TOkoyo/Kyoto 1982), Lecture Notes in Mathematics No.1016, Springer, Heidelberg (1983) 490–518. [13] M. S. Narasimhan and C. S Seshadri, Stable and unitary bundles on a compact Riemann surface, Ann. of Math. 82 (1965) 540–567. 216

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[14] M. S. Narasimhan and G. Trautmann, Compactification of MP3 (0,2) and Poncelet pairs of conics, Pacific J. Math. 145 (1990) 255–365. ¨ [15] L. Schlesinger, Uber eine Klasse von differentialsystemen beliebiger Ordnung mit festen Kritishcher Punkten, J. f¨ur Math. 141 (1912) 96–145. [16] R. L. E. Schwarzenberger, Vector bundles on the projective plane, Proc. London Math. Soc. 11 (1961) 623–640. [17] A. Weil, The field of definition of a variety, American J. Math. 78 (1956) 509–524.

Mathematics Institute University of Warwick Coventry CV4 7AL U. K.

217

Scalar conservation laws with boundary condition K. T. Joseph

1 Introduction 187

Many of the balance lawas in Physics are conservation laws. We consider scalar conservation laws in a single space variable, ut ` f puq x “ 0.

(1.1)

On the flux function f puq, we assume either f 2 puq ą 0 and lim

|u|Ñ8

f puq “8 |u|

(H1 )

or “ ‰ f puq “ log aeu ` be´u ,

(H2 )

where a and b are positive constants such that a ` b “ 1. An important 2 special case is the Burgers equations i.e., when f puq “ u2 . Initial value problem for (1.1) is to find upx, tq satisfying (1.1) and the initial data upx, 0q “ u0 pxq. (1.2) It is well known that (see Lax [8]) solution of (1.1) in the classical sense develop singularities after a finite time, no matter how smooth the initial data u0 pxq is and cannot be continued as a regular solution. The can be continued however as a solutions in weak sense. However, weak solutions of (1.1) are not determined uniquely by their initial values. Therefore, some additional principle is needed for prefering the physical solution to others. One such condition is (See Lax [9]), upx ` 0, tq ď upx ´ 0, tq. 218

(1.3)

Scalar conservation laws with boundary condition

219

This condition is called entropy condition. 188 Existence and uniqueness of weak solution of (1.1) and (1.2) satsifying the entropy condition (1.3) is well known (see Hopf [2], Lax [9], Olenik [14], Kruskov [7] and Quinn [13]). Hopf [2] derived an explicit 2 for the solution when f puq “ u2 and Lax [9] extended this formula for general convex f puq. Let f ˚ puq be the convex dual of f puq i.e., f ˚ puq “ maxruθ ´ f pθqs, θ

U0 pxq “

ż1

u0 pyqdy

(1.4) (1.5)

0

and

˙ „ ˆ x´y ˚ . u0 pyq ` t f Upx, tq “ min ´8ăyă8 t

(1.6)

For each fixed px, tq, there may be several minimisers y0 px, tq for (1.6), define ´ y` 0 px, tq “ maxty0 px, tqu, y0 px, tq “ minty0 px, tqu.

(1.7)

` Lax [9] proved that, for each fixed t ą 0, y´ 0 p., tq and y0 p., tq are left continiuous and righrt continuous respectively, and both are continuous except on a common denumerable set of points of x and at the points of continuity ´ y` 0 px, tq “ y0 px, tq.

Define

˙ x ´ y0 px, tq upx, t “ p f q t ˜ ¸ ˘ x ´ y px, tq 0 upx˘, tq “ p f ˚ q1 . t ˚ 1

ˆ

(1.8)

(1.9)

Clearly upx, tq is well defined A.e. px, tq and upx˘, tq is well defined for all px, tq Lax [9] proved the following theorem. 219

220

K. T. Joseph

Theorem 1. upx, tq defined by (1.8) and (1.9) is the weak solution of (1.1) and (1.2) which satisfies the entropy condition (1.3). Let us consider the mixed initial boundary value problem for (1.1) in x ą 0, t ą 0. We prescribe the initial data. upx, 0q “ u0 pxq, x ą 0.

(1.10)

It follows from the work of Bardos et al [1] that we really cannot impose such a boundary condition up0, tq “ λptq 189

(1.11)

at X “ o, arbitarily and hope to have a solution. Bardos et al studied this problem in several space variables by vanishing viscosity method. For one space variable their formulation is as follows. sup kPIpup0`,tq,λptqq

tsgnpup0`, tq ´ kqp f pup0`, tqq ´ f pkqqu “ 0 a.e t ą 0

(1.12) where the closed interval Ipu, λq is defined by Ipu, λq “ rminpu, λq, maxpu, λqs. Under the assumption f 2 puq ą 0, (1.12) is equivalent to saying (see Lefloch [10]) $ either ’ ’ ’ &up0`, tq “ λ` ptq (1.13) ’ or ’ ’ % 1 f pup0 ` tqq ď 0 and f pup0`, tqq ě f pλ` ptqq where

λ` ptq “ maxtλptq, u˚ u,

(1.14)

and u˚ is th solution of f 1 puq “ 0. There exists only the solution u˚ , because of the strict convexity of f puq. Definition. Let u0 pxq be in L8 p0, 8q and λptq is continuous, by a solution of (1.1) (1.10) and (1.11) we mean a solution in the sense of BardosLeroux and Nedelec. That is bounded measurable function upx, tq in 220

Scalar conservation laws with boundary condition x ě 0, t ą 0 such that ż8ż8 0

0

puφt ` f puqφ x qdxdt “ 0

221

(1.15)

for all test functions φpx, tq P C08 p0, 8q ˆ p0, 8q, upx, 0q satisfies (1.10) a.e. x ą 0, upx ` tq and upx´, tq satisfy (1.3) for x ą 0 and up0`, tq satisfies (1.13). Bardos et al [1] proved the existence and uniqueness of solution of (1.1), (1.10) and (1.11) (see also Lefloch [10]). We are interested in extending Lax formula (1.8) for the solution, which contains solution of a variational inequality. This cariational inequality is not solvable explicit. In a series of papers Joseph [3], Joseph and Veerappa Gowda [4, 5], an explicit formuala is derived for the solution of (1.1), (1.10) and (1.11), generalizing theorem 1. The case os two boundaries is considered in [6]. Before teh statement of our main theorem we introduce some notations. For each fixed px, y, tq, x ě 0, t ą 0 and α ą 0, Cα px, y, tq denotes the following class of paths pβpsq, sq in the quarter plane D “ tpz, sq : z ě 0, s ě 0u. Each pathe connects the point py, 0q to px, tq and is of the form z “ βpsq where βpsq is piecewise linear function with one line of three straight lines of possible shapes shown in Figure, share the absolute value of slope of each straight line is ď α.

221

190

222

K. T. Joseph

Let u0 pxq P L8 p0, 8q and λptq is continuous. Let λ` ptq be defined by (1.14) and let # 8 if f puq satisfies pH1 q α“ (1.16) 1 if f puq satisfies pH2 q For each fixed px, y, tq, x ě 0, y ě 0, define Qpbx, by, btq “

min

βǫCα px,y,tq

„ ż ´

`

ts:βpsq“0u

f pλ psqqds `

ż

f

‹

ts:βpsqą0u

ˆ

ds Bs

˙

ds

(1.17)

It can be shown that Qpx, y, tq is Lipschitz continous function. Let Q1 px, y, tq “ and

„ż y

B Qpx, y, tq Bx

(1.18)

u0 pzqdz ` Qpx, y, tq .

(1.19)

` y´ 0 px, tq “ minty0 px, tqu, y0 px, tq “ maxty0 px, tqu

(1.20)

Upx, tq “ min

0ďă8

0

For each fixed px, tq there may be several minimisers for (1.19). Define

191

It can be shown that (see [5]), for each fixed t ą 0, y´ 0 p., tq and are left continuous and right continuous respectively, and both are continuous except on a comman denumerable set of points of x and at the points of continuity y` 0 p., tq

´ y` 0 px, tq “ y0 px, tq

Define upx, tq “ Q1 px, y0 px, tq, tq and upx˘, tq “

Q1 px, y˘ 0 px, tq, tq

(1.21) (1.22)

Clearly upx, tq is well defined a.e x ą 0, t ą 0 and upx˘, tq is well defined for all x ą 0, t ą 0. Our main result is the following theorem. 222

223

Scalar conservation laws with boundary condition

Theorem 2. Let upx, tq be given by (1.21) and (1.22), then it is the solution of (1.1) (1.10), (1.13) and (1.3). Here we remark that viscosity solution of Hamilton Jacobi equation with Neumann boundary condition (see Lions [11, 12]). $ ’ &Ut ` f pU x q “ 0 Upx, 0q “ U0 pxq (1.23) ’ % “U x p0, tq “ λptq”

is closely related to our problem. In fact the proof of theorem 2 shows the following result. Theorem 3. The function Upx, tq defined by Upx, tq “ min rU0 pyq ` Qpx, y, tqs 0ďyď8

(1.24)

is a viscosity solution of (1.23). Here Qpx, y, tq is defined the same way a (1.17). We arrived at these results,by first working out two examples namely the Burgers equation [3] and the Lax’s equation [4].

2 Burgers Equation In this section, we conside the Burgers equation in x ą 0, t ą 0 ˆ 2˙ $ ǫ ’ ’ut ` u “ u xx , ’ & 2 x 2 Let us define U ǫ px, tq “ ´

’ upx, 0q “ u0 pxq, ’ ’ % up0, tq “ λptq,

ż8 x

upy, tqdy, U0 pyq “ ´ 223

ż8 0

u0 pyqdy.

(2.1)

(2.2)

224 192

K. T. Joseph

Then (2.1) becomes $ ˆ 2˙ Ux ’ ’ U ` “ 2ǫ U xx , ’ & t 2 ’ Upx, 0q “ U0 pxq, ’ ’ % U x p0, tq “ λptq.

(2.3)

by the Hopf-Cole transformation

1

V “ e´ ǫ U,

(2.4)

We can linearize the problem (2.3) $ V “ 2ǫ V xx , ’ & t ǫV x p0, tq ` λptqVp0, tq “ 0, ’ % 1 Vpx, 0q “ e´ ǫ U0 pxq .

(2.5)

When λptq “ λ, a constant, the solution of (2.5) can be explicitly written down: ﬁ ﬁ » » px ´ yq2 ﬂ –U0 pyq` ż 8 ´1 ǫ ﬃ — 1 2t ǫ ﬃ` — V px, tq “ dy e ﬂ – p2πtǫq1{2 0 » px ´1 – ǫ U0 pyq`

ż8

ﬁ

´ yq2 ﬂ 2t

2pλ{ǫq ` e dy ` p2πtǫq1{2 0 » ﬁ py ` zq2 1 ﬂ ż 8 ż 8 ´ –λpy´zq`U0 pyq` ǫ 2t dzdy. e 0

(2.6)

y

From (2.4), we have U ǫ px, tq “ ´ǫ logpV ǫ px, tqq.

(2.7)

Substituting (2.6) in (2.7), we have an explicit formula for U ǫ px, tq. Using the method of stationary phase one can show that the limǫÑ0 U ǫ 224

Scalar conservation laws with boundary condition

225

px, tq “ Upx, tq exists and is given by (1.24). The general variable boundary data λt can be treated by using a comparison theorem and some elementary convex analysis. The details can be found in [3].

3 Lax’s equation 193

In this section, we consider the Lax’s equation ` “ ‰ ˘ ut ` log aeu ` be´u x “ 0,

(3.1)

In x ą 0, t ą 0. In the case of no boundary Lax [9] studied this example by a difference scheme. We consider the case with boundary. Let unk » upk∆, n∆q, k “ 0, 1, 2, . . . , n “ 0, 1, 2, . . .

(3.2)

∆ being the mesh size. Let u∆ px, tq be the approximate solution defined by “ n ‰ $ n`1 n n n n “ u ` gpu , u q ´ gpu , u q u ’ k k´1 k k k` & k u0 “ u0 pkp∆qq ’ k % un0 “ λpn∆q

(3.3)

where the numerical flux gpu, vq is given by “ ‰ gpu, vq “ log aeu ` be´v .

Let Ukn “ ´

8 ÿ

j“k

unj , Ukk “ ´

8 ÿ

j“k

u0 p j∆q, Ukn “ log Vkn ,

then from (3.3), we get $ n`1 n n ’ “ aVk`1 ` bVk´1 , n “ 1, 2, . . . , k “ 1, 2, 3, . . . V ’ & k 0 Vk0 “ e´Uk , k “ 0, 1, 2, ’ ’ % n V0 “ eλpnq V1n , n “ 1, 2, . . . 225

(3.4)

(3.5)

(3.6)

226

K. T. Joseph

when λ “ λptq, a constant, an explicit formula can be obtained for the solution of (3.6) namely

194

where

$ n ˆ ˙ ÿ n q n´q 0 ’ pn´kqλ ’ e a b V2n´2q ’ ’ ’ q ’ q“0 ’ ’ ’ ’ n´k´1 & ÿ n ` e jλ S n,k` j , if n ą k Vk “ ’ ’ j“0 ’ ’ ’ ˆ ˙ n ’ ÿ ’ n q n´q 0 ’ ’ a b Vn`k´2q , if n ď k ’ % q q“0

S n,k “

and

r ÿ

q“0

(3.7)

´ ¯ 0 0 n q n´q Vn`k´2q ´ eλ Vn`k`1´2q Cq,k a b

$ ’ & n ` k ` ´1 if pn ` kq is odd r “ n ` k2` ´2 ’ % if pn ` kq is even #` ˘ 2 n , if q ď k ´ 1 n q ˘ ` n ˘ ` Cq,k “ n q ´ q´k , if q ě k ˆ ˙ n n! “ j j!pn ´ jq!

By retracing the transformation and using (3.7), we get an explicit ş8 (3.5), ∆ ∆ formula for U px, tq “ ´ x u py, tqdy. Using Stirling’s asymptotic formula one can study the limit, lim∆Ñ0 U ∆ px, tq “ Upx, tq and show that Upx, tq is given by (1.24). As in the case the Burgers Equation here agin once can prove a comparison theorem and with the help of this comparison theorem general variable λptq can be treated. The details are omitted and can be found in [4]. 226

Scalar conservation laws with boundary condition

227

4 Proof of Main theorem The examples of last two section suggest the formula for the case of general convex f puq. Following Lax [9], we introduce ş8 Q1 px, y, tqe´Nru0 pyq`Qpx,y,tqs dy uN px, tq “ 0 ş8 (4.1) ´NrU0 pyq`Qpx,y,tqs dy 0 e ş8 f pQ1 px, y, tqqe´NrU0 pyq`Qpx,y,tqs dy (4.2) fN px, tq “ 0 ş8 ´NrU0 pyq`qpx,y,tqs dy 0 e ż8 e´NrU0 pyq`Qpx,y,tqsdy . (4.3) VN px, tq “ 0

1 log VN . (4.4) N where U0 pxq, Qpx, y, tq,Q1 px, y, tq are defined in Sec 1. It is clear that # limNÑ8 uN px, tq “ Q1 px, y0 px, tq, tq a.epx, tq (4.5) limNÑ8 fN px, tq “ f rQ1 px, y0 px, tq, tqs a.epx, tq U N px, tq “ ´

where y0 px, tq minimises (1.19) and

195

lim U N px, tq “ Upx, tq,

(4.6)

BU “ Q1 px, y0 px, tq, tq. Bx

(4.7)

NÑ8

and It can be shown that

ż8ż8 0

puN φt ` fN qdxdt 0 C08 p0, 8q ˆ p0, 8q.

“0

(4.8)

for all test function φ P Let N Ñ 8 and use (4.5) we get from (4.8) upx, tq “ Q1 px, y0 px, tq, tq

(4.9)

solves (1.1) in distribution. It can be shown that upx, tq defined by (4.9) satisfies initial condition (1.10) boundary condition (1.13) and entropy condition (1.3). The details can be found in [5]. 227

228

K. T. Joseph

References [1] Bardos, C. Leroux, A.Y. and Nedelec, J.C., First order quasilinear equations with boundary conditions, Comm. Partial Differential equations 4 (1979) 1017–1034. [2] Hopf, E. The partial differential equationut uu x “ µU xx , Comm. Pure and Appl. Math 13 (1950) 201–230. [3] Joseph, K.T. Burgers equation in the quarter plane, a formula for the weak limit, Comm. Pure and Appl. Math 41 (1988) 133–149. [4] Joseph, K. T. and Veerappa Gowda, G. D., Explicit-formula for the limit of a difference approximation, Duke Math. J. 61 (1990) 369–393. [5] Joseph, K. T. and Veerappa Gowda, G. D., Explicit-formula for the solution of convex conservation laws wit boundary condition, Duke Math. J. 62 (1991) 401–416. [6] Joseph, K. T. and Veerappa Gowda, G.D., Solution of convex conservation laws is a strip, Proc Ind. Acad. Sci, 102, 1,(1992) 29–47. [7] Kruskov, S. First order quasilinear equations with several space variables, Math. USSR Sb, 10 (1970) 217–273. 196

[8] Lax, P.D., Weak solutions of non linear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math 7 (1954) 159–193. [9] Lax, P. D: Hyperbolic Systems of Conservation laws II, Comm. Pure Appl. Math. 10 (1957) 537–566. [10] Lefloch, P.,Explicit-formula for Scalar non-linear Conservation laws with boundary condition, Math Methods in Appl. Sci 10 (1988) 265–287. [11] Lions, P.L., Generalized solutions of Hamilton-Jacobi equation, Pitman Lecture Notes, Pitman, London (1982). 228

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[12] Lions, P.L., Neumann type boundary conditions for HamiltonJacobi equations, Duke Math. J. 52 (1985) 793–820. [13] Quinn, B., Solutions with shocks: an example of L1 - contraction semi-group, Comm. Pure Appl. Math. 24 (1971) 125–132. [14] Oleinik, O.,Discontinuous solutions of non-linear differential equation, Usp. Mat. Nauk, (N.S), 12 (1957) 3–73; English transl. in AMS Transl. Ser. 2, 26 (1957) 95–172.

TIFR Centre P.B. No.1234 IISc. Campus Bangalore 560 012

229

Bases for Quantum Demazure modules-I By V. Lakshmibai* (Dedicated to professores M. S. Narasimhan and C.S. Seshadri on their 60th birthdays)

1 Introduction 197

Let g be a semi-simple Lie algebra over Q of rank n. Let U be the quantized enveloping algebra of g as constructed by Drinfeld (cf [D]) and Jimbo (cf [J]). This is an algebra over Qpvq (v being a parameter) which specializes to Upgq for v “ 1, upgq being the universal enveloping algebra of g. This algebra has agenerators Ei , Fi , Ki , 1 ď i ď n, which satisfy the quantum Chevalley and Serre relations (cf [L1]). Let A “ Zrv, v´1 s and U A˘ be the A-sumbalagebra of U generated by Eir (resp. Fir ), 1 ď i ď n, r P Z` , (here Eir , Fir are the quantum divided powers (cf [J])). Let d “ pd1 , ldot, dn q P pZ` qn and Vd be the simple U-module with a non-zero vector e such that Ei e “ 0, Ki e “ v´di e (recall that Vd is unique up to isomorphism). Let us denote Vd by just V. Let W be the Weyl group of g. Let w P W and let w “ si1 . . . sir be a reduced expression for w. Let pa q ´ Uw,A denote the A-submodule of U spanned by Fi1 1 . . . Fiarr , ai P Z` . ´ We observe ([L4]) that Uw,4 depends only on w and not onth reduced ´ expression chosen. For w P W, let Vw,A “ Uw,A e. We shall refer to Vw,A as the Quantum Demazure module associated with w. Let w0 be the unique element in W of maximal lenght. Then Vw0 ,A is simply U A´ e. In the sequel, we shall denote Vw0 ,A by just VA . Let g “ sℓ (3). In this paper, we construct an A-basis for VA , which is compatible with tVw,A , w P Wu. The construction is done using the * Partially

supported by NSF Grant DMs 9103129 and alos by the Faculty Development Fund of Northeastern University.

230

Bases for Quantum Demazure modules-I

231

configuration of Schubert varieties in the Flag variety G{B. Let Id “ µ0 ă µ1 ă µ2 ă µ3 “ w0 be a chain in Wp“ S 3 q. Let µi´1 “ sβi µi for some positive root βi . Let npµi´1 , µi q “ pµi´1 pλq, β˚i q, where λ “ 198 d1 ω1 ` d2 ω2 , ω1 and ω2 being the fundamental weights of sℓp3q. Let denote npµi´1 , µi q by just ni . Let C “ tpµ0 , µ1 , µ2 , µ3 ; m1 , m2 , m3 q : 1 ě mn11 ě mn22 ě mn33 ě 0u. Given c “ pµ0 , µ1 , µ2 , µ3 ; m1 , m2 , m3 q, Let τc “ µr , where r is the largest integer such that mr ‰ 0. Given two elements c1 “ tpµ0 , µ1 , µ2 , µ3 ; m1 , m2 , m3 qu, c2 “ tλ0 , λ1 , λ2 , λ3 ; p1 , p2 , p3 u in C. let us denote c1 „ c2 , if

mi npµi´1 ,µi q

(resp.

pi ) npλi´1 ,λi q

by just ai (resp. bi ). We say

(1) ai “ bi (2) either (a) a1 “ a2 ą a3 , µ2 “ λ2 or (b) a1 ą a2 “ a3 , µ1 “ λ1 or (c) a1 “ a2 “ a3 We shall denote C{ „ by C, For θ P C, we shall denote τθ “ τθ, c being a representative for θ (note that τθ is well-defined). We have (Theorems 6.7 and 7.2) Theorem. VA has an A-basis Bd “ tvθ , θ P Cu where vθ “ Dθ e, Dθ 1 pn q pnrq being a monomial in Fi s of the form Fi1 1 . . . fir . Further, for w P W, tvθ |w ě τθ u is an A-basis for Vw,A . Let Bd denote Lusztig’s canonical basis for VA (cf [L2]). It turns out that the transition matrix from Bd to Bd is upper triangular. We also give a conjectural A-basis Bd for VA for g of other types. An element in Bd is pn q pnrq again of the form Fi1 1 ¨ ¨ ¨ Fir e. We conjecture that the trasition matrix 231

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from Bd to Bd is upper triangular. The sections are organized as follows. In §2, we recall some results pertaining to the configuration of Schubert varieties in G{B. In §3, we describe a conjectural A-basis for VA (and also for Vw,A ). In §4, we study C in detail and constuct Bd and Bd . In §6 and §7, we prove the results for G “ S Lp3q. in §8, an Appendix, we have explicitly established a bijection between the elements of C (the indexing set for Bd ) and the classical standard Young tableaux on S Lp3q of type pd1 , d2 q. The author would like to express her gratitude to SPIC Science Foundation for the hospitality extended to het during her stay (Jan-Feb, 1992) there, when research pertatining to this paper was carried out.

2 Preliminaries 199

Let G be a semi-simple, simply connected Chevalley group defined over a field k. Let T be a maximal k-split torus, B a Borel subgroup, B Ą T . Let W be the Weyl group, and R the root system of G relative to T . Let R` (resp. S ) be the system of positive (resp. simple) roots of G relative to B. For w P W, let Xpwq “ BwBpmod Bq be tghe Schubert variety in G{B associated with w. Definition 2.1. Let Xpϕq be a Schubert divisor in Xpτq. We say that Xpϕq is moving divisor in Xpτq moved by the simple root α, if ϕ “ sα τ. Lemma 2.2 (Cf [LS]). Let Xpϕq be a moving divisor in Xpτq moved by α. Let Xpwq be a Schubert subvariety in Xpτq. Then either (i) Xpwq Ď Xpϕq or (ii) Xpwq “ Xpsα w1 q for some Xpw1 q Ď Xpϕq. Definition 2.3. Let λ be a dominant integral weight of G. Let Xpwq be a divisor on Xpτq. Let w “ sβ τ, for some β P R` . We define mλ pw, τq as the non-negative integer mλ pw, τq “ pwpλq, β˚ qp“ ´pτpλq, β˚ qq, and call it the lambda multiplicity of Xpwq in Xpτq. (Here (,) is aW-invariant scalar product on HompT, Gm q). 232

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233

Lemma 2.4. Let Xpϕq be a moving divisor in Xpτq moved by α. Let Xpwq be divisor in Xpτq. Let w “ sβ τ, θ “ sα ϕ, where β, γ P R` (note that γ “ sα pβqq. Let m “ mλ pw, τq, s “ mλ pθ, wq, p “ mλ pϕ, tauq, r “ pβ, α˚ q. Then (a) mλ pθ, ϕq “ m (b) p “ s ` mr Proof. (a) mλ pθ, ϕq “ pθpλq, γ˚ q “ pθpλq, psα θpλqq, β˚ q “ pwpλq, β˚ q “ m (b) Now sβ sα θpλq “ sα sγ θpλq implies that sα ` mβ “ mγ ` pα. Hence pp´ sqα “ mβ´mγ “ mpβ´rαq “ mrα. Hence p “ s`mr. Lemma 2.5. Let Xpθq be a divisor in Xpτq. Let θ “ sβ τ, where β P R` . 200 Let β be non-simple, say β “ Σci αi , αi P S . Then for at least one i with ci ‰ 0, we have pτpλq, α˚i q ă 0. Proof. Let m “ mλ pθ, τq. Then pτpλq, β˚ q “ ´m ă 0. The assertion follows from this. 2.6 Lexicographic shellability Given a finite partially ordered setH which is graded (i.e., which has an unique maximal and an unique minimal element and in which all maximal chains m i.e., maximal totally ordered subsets of H, have the same length), the lexicographic shellability of H (cf [B-W]) consists in labelling the maximal chains m in H, say λpmq “ pλ1 pmq, λ2 pmq, . . . , λr pmqq (here r in the length of any maximal chain in H), where λi pmq belong to some partially ordered set Ω, in such a way that the following two conditions hold: (L1) If two maximal chains m and m1 coincide along their first d edges, for some d, 1 ě d ě r, then λi pmq “ λi pm1 q, 1 ď i ď d. 233

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(L2) For any interval rx, ysp“ τ P H : X ě τ ě yq, together with a chain c going down from the unique maximal element in H to y, there is a unique maximal chain m0 in rx, ys whose label is increasing (i.e. λ1 pm0 q ď λ2 pm0 q ď ¨ ¨ ¨ λt pm0 q, t being the lenght of any maximal chain in rx, ys) and if m is any other maximal chain in rx, ys, then λpm0 q is lexicographically , λpmq (here, the label for any chain m in rx, ys is induced from the maximal chain of H consisting of c, followed by m0 , followed by an arbitrary path from x to the unique minimal element of H.) Theorem 2.7. (cf [B-W]) the Bruhat order of a Coxeter group is lexicographic shellable. 2.8 Labelling of maximal chains in rx, ys for H “ W We fix a reduced expression of w0 ,the element of the maximal length in W and label the maximal chains in X as in [B-W](with respect to this fixed reduced expression for w0 ). Let m be maximal chain in rx, ys. Let c be the (unique) chain from w0 to y whose label is increasing. We take the label for m as the induced by maximal chain in W consisting of c, followed by m, followed by an arbitrary path from x to Id.

3 A conjectural Bruhat-order compatible A-basis for VA 201

3.1 Let g “ LiepGq. Let U, A, U A˘ , Ei , Fi , Ki , Vd , VA , Vw,A etc. be as in §1. Let λ “ Σdi ωi , ωi being the fundamental weights of G. We shall index that set of simple roots of G as in [B]. Let c “ tµ0 , µ1 , . . . , µr u be a chain in W, i.e., ℓpµi q “ ℓpµi´1 q ` 1 (if dt “ 0 for t “ ii , . . . , i s , then we shall work with W Q , the set of minimal representatives of WQ in W, WQ being the subgroup of W generated by the set of simple reflections tst , t “ i1 . . . , i s u). Let µi´1 “ sβi µi , βi P R` . Let mλ pµi´1 , µi q “ mi . We set ℓpcq “ r, and call it the length of c. Definition 3.2. A chain c is called simple (resp. non-simple if all (resp. some) β1i s are simple (resp. non-simple)). 234

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235

Definition 3.3. By a weighted chain, we shall mean (c, n) where c “ tµ0 , . . . , µr u is a chain and n “ tn1 , . . . , nr u, ni P Z` . Definition 3.4. A weighted chain (c, c, n) is said to be admissible if 1 ě n2 nr n1 m1 ě m2 ě ¨ ¨ ¨ ě mr ě 0. Lemma 3.5. Let Xpϕq be a moving divisor in Xpτq moved by the simple root α. Let c “ tµ0 , µ1 , . . . µr “ τu be a chain, and let µi´1 “ sβ µi , βi P R` . Further let βr ‰ α, and βi P S , i ‰ r (note that we allow βr to be non-simple). Then either (1) βi “ α, for some i, (or) (2) βi ‰ α, 1 ď i ď r, in which case µi ą sα µi , and sα µi ă ϕ, 0 ď i ď r Proof. Let βi ‰ α, 1 ď i ď r. We shall now show that µi ą sα µi , 0 ď i ď r. For i “ r, this is clear (since sα µr “ ϕ ă τ). For i “ r ´ 1,this follows from Lemma 2.2 We have pµr´2 pλq, α˚ q “ pµr´1 pλq ` mr´1 βr´1 , α˚ q ă 0, since pµr´1 pλq, α˚ q ă 0, and pβr´1 , α˚ q ď 0. (note that µr´1 ą sα µr´1 ñ pµr´1 pλq, α˚ q ă 0 ,and that pβr´1 , α˚ q ď 0, since βi P S , 1 ď i ď r ´ 1). Hence µr´2 ą sα µr´2 . In a similar way one concludes µi ą sα µi . The assertion that sα µi is ă ϕ follows Lemma 2.2. 202

3.6 Let pc, nq be an admissible weighted chain. With notations and assumptions as in Lemma 3.5, we define as admissible weighted chain psα pcq, sα pnqq as follows: Case 1. Let βi ‰ α, 1 ď i ď r. We set sα pcq “ tsα µ0 , sα µ1 , . . . , sα µr “ ϕu

sα pnq “ n

(Note that mλ psα µi´1 q, sα µi q “ mλ pµi´1 , µi q (Lemma 2.4 (a)) and hence (sα pcq, sα pnqq is admissible). 235

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Case 2. Let βi “ α, from some i, 1 ď i ď r. Let t be the largest integer, 1 ď t ď r such that βt “ α. We set sα pcq “ tµ0 , µ, ¨ ¨ ¨ , µr´1 “ sα µt , sα µt , sα µt`1 , ¨ ¨ ¨ , sα µr “ ϕu and we define sα pnq “ tn1i , ¨ ¨ ¨ , n1r´1 u by n1i “ ni , 1 ď i ď t ´ 1

n1i “ ni`1 , t ď i ď r ´ 1 n1r´1 n11 ě ¨ ¨ ¨ ě 1 m1 m1r´1 1 mi “ mi , 1 ď i ď t

(note that 1 ě

Case 1 (here i ď r ´ 1q.

ě 0, by the same considerations as in

´ 1, m1i p“ psα µi , sα µi`1 qq “ mi`1 , t ď

3.7 With notations and assumptions as in 3.6, we donet by pcα , nα q the admissible weighted chain, where cα “ psα pcq, µr q, nα “ psα pnq, n1r q and n1r in given as follows: Let # 0, if case 1 holds k“ t, if case 2 holds For i ě k, let γi “ sα pβi q (note that sα µi´1 “ sγi sα µi ). # mpsα µ0 , µ0 q, if case 1 holds x“ nt , if case 2 holds #ř ni pβi , α˚ q y“ ti ą k | γi ‰ βi u 203

Then n1r is given by

n1r “ x ` y n1r “ x ` y 236

Bases for Quantum Demazure modules-I

237

3.8 Let pc, nq be an admissible weighted chain. Further, let c be not simple. To pc, nq, we attach a canonical (not necessarily admissible) weighted chain pδpcq, δpnqq with δpcq simple, as follows: We preserve the above notation for c. We do the construction using induction on rp“ ℓpcqq. Starting point of induction Let c “ tpτ, wqu,where xpwq is a divisor in Xpτq, w “ sβ τ, β P R` , β non-simple. (We refer to this situation as Xpwq being a non-moving divisor in Xpτq). Let n “ pn1 q, where n1 ď mp“ mλ pw, τqq. By induction on dim Xpτq, we may suppose that Xpτq is of least dimension such that Xpτq has a non-moving divisor. Let β “ Σci αi . then at least for one i with ci ‰ 0, we have pτpλq, α˚i q ă 0 (Lemma 2.5). Let t be at least for one i with ct ‰ 0 such that pτpλq, α˚t q ă 0 (the indexing of the simple roots being as in [B]). Denote αt by just αt by just α. Let salpha w “ θ, ϕ “ sγ θ. Then γ “ sα pβq. Further by our assumption on dim Xpτq, γ P S . Hence we obtain pβ, α˚ q ą 0, say pβ, α˚ q “ r, and β “ γ ` rα. We set pδpcq, δpnqq “ tpθ, ϕ, τq; pp1 , p2 qu where p1 “ n1 , P2 “ n1 ` a with a “ mλ pθ, wq (lemma 2.4). Let now ℓpcq ą 1. Let c “ tµ0 , µ1 , . . . , µr u, µi´1 “ sβ µi , 1 ď i ď r. We may suppose that βi P S , 1 ď i ă r. For, otherwise, if i is the least integer such that βi in non-simple we may work with p∆pnq, ∆pcqq, (where ∆pcq is the chain δpµ0 , ¨ ¨ ¨ , µi q followed by tµi`1 , ¨ ¨ ¨ , µr u, and use induction on #tt, 1 ď t ď r : βt is non-simpleu. Let us denote βr by just β. Let β “ Σci αi . Since pτpλq, β˚ q ă 0, we have (Lemma 2.5, for at least one t with ct ‰ 0, pτpλq, α˚t q ă 0. Let i be the least integer such that ci ‰ 0 and pτpλq, α˚i q ă 0. Let us denote αi by just α. Let ϕ “ sα τ. We set (3.7) δpcq “ cα , δpnq “ nα 3.9 Given a simple weighted chain pc, nq (not necessarily admissible) we set pn q pn q vc,n “ Fir r ¨ ¨ ¨ Fi1 1 eµ 237

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where c “ tµ “ µ0 , . . . µr u, n “ tn1 , . . . , nr u, βt “ αit , 1 ď t ď r, and eµ is the extermal weight vector associated to µ, (Note that it τ0 “ Id ă τ1 ă ¨ ¨ ¨ ă τr “ µ is anu simple chain from Id to µ, pn q pn q and τi´1 “ sβ τi , βi P S , 1 ď i ď r, then eµ “ Fβr r . . . Fβ1 1 e, where ni “ mλ pτi´1 , τi q (2.3)) 204

3.10 Let pc, nq be admissible. Let us denote the unequal values in t mn11 , ¨ ¨ ¨ , mnrr u by a1 , ¨ ¨ ¨ , a s so that 1 ě a1 ą a2 ą ¨ ¨ ¨ ą a s ě 0. Let i0 , ¨ ¨ ¨ , i s be defined by nj i0 “ 0, i s “ r, “ at , it´1 ` 1 ď j ď it mj We set Dc,n “ tpa1 , ¨ ¨ ¨ , a s q; pµi0 , ¨ ¨ ¨ , µis qu

Definition 3.11. Let pc, nq, pc1 , n1 q be two admissible weighted chains. Let Dc,n„ pc1 , n1 q, if s “ t, and at “ a1t ,it “ jt , µit “ τ jt , 0 ď t ď s 3.12 Given µ, τ P W, µ ă τ, we shall label the chain in rµτs as in 2.8. Let C “ tall admissible weighted chainsu, and C “ C{„ . Given x P C, let S x “ tpc, nq P C : pc, nq is a representative of x and c is simpleu N x “ tpc, nq P C : pc, nq is a representative of x and c in non-simpleu Let us define xmin P C as follows.

Case 1. S x ‰ φ. We set xmin “ pc0 , n0 q where pc0 , n0 q is lexicographically the least in S x . Case 2. S x ‰ φ. In this case we set xmin “ pc0 , n0 q where pc0 , n0 q is lexicographically the least in N x . 238

Bases for Quantum Demazure modules-I

239

3.13 Let x P C. Let xmin “ pc0 , n0 q. Define v x P VA as follows: Case 1. S x ‰ φ. We set

v x “ vc0 ,n0

Case 2. S x “ φ. We set v x “ vδpc0 q,δpn0 q 3.14 Let x P C. Let pc, nq be a representative of x. Let c “ pµ0 , ¨ ¨ ¨ , µr q. We define τ x “ µ s , where is s is the largest integer such that n x ‰ 0 (Note that τ x is well-defined). Conjectures 3.15.

205

(1) tv x , x P Cu is an A-basis for VA (2) tv x : w ě τ x u is an A-basis for A, w (3) Let Bd denote tv x , x P Cu, and Bd , Lusztig’s canonical basis for VA ([L2]). The transition matrix from Bd to Bd is upper triangular.

4 The case G = SL p3q 4.1 For the rest of the paper we shall suppose that G “ S Lp3q. Let us denote that elements of W by tτi , φi , i “ 0, 1, 2, 3, j “ 1, 2u, where τ0 “ Id, τ1 “ s1 , τ2 “ s2 s1 , τ3 “ s1 s2 s1 , ϕ2 “ s1 s2 . We shall label the maximal chains in W with respect to teh reduced expression s1 s2 s1 of w0 ([B-W]). 4.2 Let d “ pd1 , d2 q and λ “ d1 ω1 ` d1 ω2 . We shall suppose that d1 , d2 are both non-zero (If d1 “ 0 for instance,then we work with tId, s2 , s1 s2 u, the set of minimal representatives of W p p“ ts1 , Iduq in W). Also, for simplicity of notation, we shall denote d1 by m and d2 by n. 239

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4.3 Given a pair pϕ, , τq such that Xpϕq is a divisor in Xpτq, let us denote mλ pϕ, τq bu just mpϕ, τq. We have $ ’ &m, if pϕ, τq “ pτ0 , τ1 q, pϕ1 , τ2 q or pϕ2 , t3 q mpϕ, τq “ m ` n, if pϕ, τq “ pτ1 , τ2 q or pϕ1 , ϕ2 q ’ % n, if pϕ, τq “ pτ0 , ϕ1 q, pτ1 , ϕ2 q or pτ2 , τ3 q 4.4 We shall denote an admissible weighted chain c “ pµ0 , µ1 , µ2 , µ3 q, n “ pn1 , n2 , n3 q, where if ni “ 0, i being the least such integer, then c is to be understood as the chain pν0 , ¨ ¨ ¨ , µi´1 q. For instance, if n3 “ 0 and n1 , n2 are non-zero then c “ pµ0 , µ1 , µ2 q. If n1 “ 0 “ n2 “ n3 , we shall call c a trivial chain consisting of just µ0 .

4.5 We have four types of admissible weighted chains given as follows.

206

n3 n

Type I: tpτ0 , τ1 , τ2 , τ3 q, pn1 , n2 , n3 q : 1 ě

n1 m

ď

n2 m`n

Type II: tpτ0 , ϕ1 , τ1 , τ3 q, pn1 , n2 , n3 q : 1 ě

n1 n

ď

n2 m

ě

n3 n

ě 0u

Type III: tpτ0 , τ1 , ϕ2 , τ3 q, pn1 , n2 , n3 q : 1 ě

n1 m

ď

n2 n

ě

n3 m

ě 0u

Type IV: tpτ0 , ϕ1 , ϕ2 , τ3 q, pn1 , n2 , n3 q : 1 ě

n1 n

ď

n2 m`n

ě

ě

n3 m

ě 0u

ě 0u

4.6 Given ∆1 “ ppµ0 , µ1 , µ2 , µ3 q, pn1 , n2 , n3 qq, ∆2 “ ppλ0 , λ1 , λ2 , λ3 qpp1 , p2 , p3 qq in C, let use denote have ∆1 „ ∆2 , if

ni mpµi´1 ,µi q

(resp.

pi ) mpλi´1 ,λi q

(1) ai “ bi (2) (a) a1 “ a2 ą a3 , µ2 “ λ2 , or (b) a1 ą a2 “ a3 , µ1 “ λ1 , or (c) a1 “ a2 “ a3

240

by just ai (resp. bi ). We

241

Bases for Quantum Demazure modules-I

We note that if (a) holds, then λ2 “ τ2 or ϕ2 , and the equivalence can hold only between elements of either Type I and II, or Type III and IV repectively if (b) holds, then λ1 “ τ1 or ϕ1 , and the equivalence can hold only between elements of either Type I and III or Type II and IV respectively. 4.7 Let x P C. Let xmin “ pc, nq (3.12). Let c “ pµ0 , µ1 , µ2 , , µ3 q, n “ pn1 , n2 , n3 q, and ai , 1 ď i ď 3 as above. Note that if in 4.5, 2a holds, and µ2 “ τ2 presp.ϕ2 q, then pc, nq is of Type I (resp Type IV). If 2b holds, and µ1 “ τ1 presp.ϕ1 q, then pc, nq is of Type I (resp. type IV). If 2c holds, then pc, nq is of Type I. 4.8 With notation as in 4.7, the element pressed explicity as $ pn q pn q pn q ’ if F1 3 F2 2 F1 1 e, ’ ’ ’ &F pn3 q F pn1 `n2 q F pn2 q e, if 1 2 1 vx “ pn3 q pn1 `n2 q pn2 q ’ F F F ’ 2 1 2 e, if ’ ’ % pn3 q pn2 q pn1 q F2 F1 F2 e, if

v x P VA (3.13) may be exc

is of Type I

c

is of Type II

c

is of Type III

c

is of Type IV

(note that v x is external ðñ ni is either 0 or “ mi , where mi “ mλ pµi´1 , µi q). We shall denote tv x , x P Cu by Bd . 4.9 Lusztig’s canonical basis for V A . An element in VA of the form ppq pqq prq puq ptq psq F1 F2 F1 e, q ě p ` r or F2 F1 F2 e, t ě u ` s will be referred to prq

as a Lusztig element or just a L-element. We have F1 e ‰ 0 ðñ r ď m. prq

Let r ď m; then F2 e ‰ 0 ðñ q ď r ` n (using the relation pqq

prq

F2 F1 “

minpq,rq ÿ

pr´ jq

v´ j´pq´ jqpr´ jq F1

j“0

p jq

pq´ jq

Fα1 `α2 F2 p

q

q

Let now, r ď m, p ` r ď q ď r ` n; then F1 F2 F1r e ‰ 0 (by 207 pqq

prq

pqq

prq

Uq pℓ2 qq-theory since K1 pF2 F1 eq “ v´a F2 F1 e (where a “ m ` 241

242

V. Lakshmibai ppq

pqq

prq

q ´ 2r, and p ď a). Thus an L-element of the form F1 F2 F1 e (resp. puq

ptq

psq

F2 F1 F2 e) is non-zero if and only if r ď m, and q ď r ` n (resp. s ď n, t ď s ` m). Hence if Bd denotes Lusztig’s canonical basis for VA , then + # ppq pqq prq F1 F1 F1 e, p ` r ď q ď r ` n, r ď m, and Bd “ puq ptq psq F2 F1 F2 e, u ` x ď t ď s ` m, s ď n ppq

pqq

prq

prq

pqq

ppq

(here one notes that the if q “ p`r, then F1 F2 F1 e “ F2 F1 F2 e) Let p, q, r, p1 , q1 , r1 P Z` . Let # + pp, q, rq, r ď m, p ` r ď q ď r ` n, and L“ pp1 , q1 , r1 q, r1 ď n, p1 ` r1 ď q1 ď r1 ` m where pp, q, rq is identified with pp1 , q1 , r1 q, if q “ q1 “ r ` p, p “ r1 , r “ p1 (note that L is an indexing set for Bd ).

5 A bijection between L and C Lemma 5.1. Let pp, q, rq in L be such that r ď m, p ` r ď q ď r ` n. Further let mr , np , q´r n be all distinct. Then precisely one of the following holds q m`n

p n

(1) 1 ě

r m

(2) 1 ě

q´r n

ě

r m

ě

p n

(3) 1 ě

q´r n

ě

p`r m`n

ě

ě

ě

ě0 ě0 r m

ě0

Proof. We first observe that under the hypothesis that mr , np q´r n are distinct, the three cases are mutually exclusive. We now distinguish the following two cases. q Case 1. mr ă m`n q´r p`r r This implies that q´r n ą n . In this case m`n ă n , necessarily; for p`2r´q r`p g´r r pě q´r m`n ě n would imply m n q ą m , which is not possible, since q ě p ` r. Hence either

242

243

Bases for Quantum Demazure modules-I (a) 1 ě

q´r n

ě

p`r m`n

(b) 1 ě

q´r n

ě

r p`r m m`n

ě

r m

ě 0, in which case (??) holds or

ě 0, in which case (2) holds.

q The hypothesis that q ě p ` r implies that Case 2. mr ě m`n Thus in this case (1) holds.

q m`n

ě np .

5.2 Let pp, q, rq in L be such that r ď m, p ` r ě q ě r ` n. We 208 now define an element θpp, q, rq in C. Let us denote θpp, q, rq “ pc, nq, where c and n are given as follows. (a) Let mr , np , q´r n be all distinct. We set

(b) Let

r m

$ ’ &pτ0 , τ1 , τ2 , τ3 q, c “ pτ0 , ϕ1 , τ2 , τ3 q, ’ % pτ0 , ϕ1 , τ2 , τ3 q, $ ’ &pr, q, pq, n “ pq ´ r, r, pq, ’ % pq ´ r, p ` r, rq,

“

q´r n

if if if

(1) of (2) of (3) of

if (1) if (2) if (3)

of of of

(5.1) (5.1) (5.1)

holds holds holds

(5.1) holds (5.1) holds (5.1) holds

‰ np .

Then (3) of 5.1 cannot hold, and the cases (1) and (2) of 5.1 coincide. We set c “ pτ0 , τ1 , τ2 , τ3 q, n “ pr, q, pq (c) Let

r m

“

p n

‰

q´r n .

Then (1) of 5.1 cannot hold, and (2) and (3) of 5.1 coincide. We set c “ pτ0 , ϕ1 , ϕ2 , τ3 q, n “ pq ´ r, p ` r, rq 243

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(d) Let

p n

“

q´r n

“

p n

r m.

‰

This implies that q “ r ` p. In this case (2) of 5.1 cannot hold, and (3) are mutually exclusive. We set # pτ0 , τ1 , τ2 , τ3 q, if (1) of 5.1 holds c“ pτ0 , ϕ1 , ϕ2 , τ3 q if (3) of 5.1 holds # pr, q, pq, if (1) of 5.1 holds n“ pq ´ r, p ` r, rq, if (3) of 5.1 holds (e) Let

r m

“

p`r n`m .

q “ This implies that m`n coincide, and we set

r m

“

p`r n`m .

Then all three cases of 5.1

c “ pτ0 , τ1 , τ2 , τ3 q, n “ pr, q, pq 209

Remark 5.3. We observe that in all of the cases (a) through (e) above, pc, nq is of type I, II or IV. Also, if x “ pc, nq, then it is easily seen that xmin “ pc, nq. Lemma 5.4. Let pp1 , q1 , r1 q in L be such that r1 ď n, p1 `r1 ď q1 ď r1 `m. 1 1 p1 1 Further, let rn , q ´r m m be all distinct. Then precisely one of the following holds. (1)

r1 n

ě q1 m ` n ě

(2) 1 ě (3) 1 ě

q1 ´r1 m q1 ´r1 m

ě ě

r1 n

p1 m

ě

p1 `r1 m`n

ě0 p1 m

ě

ě0 r1 n

ě0

The proof is similar to that of Lemma 5.1. 5.5 Let pp1 , q1 , r1 q in L be such that r1 ď n, p1 ` r1 ď r1 ` m. We now define an element θpp1 , q1 , r1 q in C. Let us denote θpp1 , q1 , r1 q “ pc1 , n1 q, where c1 and n1 are given as follows. 244

245

Bases for Quantum Demazure modules-I (a) Let

(b) Let

r1 p1 q1 ´r1 m, m , m ,

r1 n

be all distinct. We set

$ ’ &pτ0 , ϕ1 , ϕ2 , τ3 q, if 1 c “ pτ0 , τ1 , ϕ2 , τ3 q, if ’ % pτ0 , τ1 , τ2 , τ3 q if $ 1 1 1 ’ &pr , q , p q 1 n “ pq1 ´ r1 , r1 , p1 q, ’ % 1 pq ´ r1 , p1 ` r1 , r1 q “

q1 ´r1 m

‰

p1 m.

(1) (2) (3) if if if

of of of (1) (2) (3)

5.4 holds 5.4 holds 5.4 holds of 5.4 of 5.4 of 5.4

holds holds holds

We set

c1 “ pτ0 , ϕ1 , ϕ2 , τ3 q, n1 “ pr1 , q1 , p1 q (the discussion being as in 5.2 (b)) (c) Let

r1 n

“

p1 m

‰

q1 ´r1 n

. We set

c1 “ pτ0 , τ1 , τ2 , τ3 q, n1 “ pq1 ´ r1 , p1 ` r1 , r1 q 1

1

1

1

r (d) Let pm “ q ´r m ‰ n . In this case, (1) of 5.4 cannot hold and (1) and (3) of 5.4 are mutually exclusive.

We set # pτ0 , ϕ1 , ϕ2 , ϕ3 q, if c1 “ pτ0 , τ1 , τ2 , τ3 q, if 1

1

1

1

(1) (3) 1

of 5.4 of 5.4

holds holds

p1 `r1 m`n

q1 m`n .

r (e) Let rn “ pm “ q ´r m . This implies that n “ the three cases of 5.4 coincide and we set

“

Then all

c1 “ pτ0 , τ1 , τ2 , τ3 q, n1 “ pq1 ´ r1 , p1 ` r1 , r1 q Remark 5.6. In all of the cases (a) through (e) above, (c1 , n1 ) is of Type 210 I, III or IV. Also if y “ pc1 , n1 q, the ymin “ pc1 , n1 q. 245

246

V. Lakshmibai

5.7 Let L1 “ tpp, q, rq P L : r ě m, p ` r ď q ď r ` nu

L2 “ tpp1 , q1 , r1 q P L : r1 ď n, p1 ` r1 ď q1 ď r1 ` mu An element pp1 , q1 , r1 q in L2 will be identified with the element pp, q, rq in L1 , if r1 “ p, p1 “ r, q1 “ q “ r ` p “ r1 ` p1 . When this happens, we shall express it as pp1 , q1 , r1 q „ pp, q, rq. Let θ be as in 5.2 (resp. 5.5). We observe that if pp1 , q1 , r1 q „ pp, q, rq, then θpp1 , q1 , r1 q “ θpp, q, rq. p To see this, let θpp, q, rq “ pc, nq, Then since q´r n “ n , only (d) or (e) of 5.2 can hold. If (e) of 5.2 holds, then (e) of 5.5 also holds. We have (5.2 (e), 5.5 (e)), c “ pτ0 , τ1 , τ2 , τ3 q, n “ pr, q, pq

1

c “ pτ0 , τ1 , τ2 , τ3 q, n1 “ pq1 ´ r1 , p1 ` r1 , r1 qp“ pr, q, pqq Thus pc, nq “ pc1 , n1 q. Let that (d) of 5.2 hold. We distinguish the following two cases. Case 1.

r m

ą

p n

This implies that 1 ě

r1 `p1 m`n

ą

r1 n

r m

ą

q m`n

ą

p n

ě

ě 0. Hence we get (5.2(d), 5.5(d))

p n

ě 0, and 1 ě

q1 ´r1 m

ą

c “ pτ0 , τ1 , τ2 , τ3 q, n “ pr, q, pq

1

c “ pτ0 , τ1 , τ2 , τ3 q, n1 “ pq1 ´ r1 , r1 ` p1 , r1 qp“ pr, q, pqq Thus pc, nq “ pc1 , n1 q Case 2. p1 m

p n

ą

. m.

This implies that 1 ě

p n

ą

p`r m

ą

ě 0. Hence we get (5.2, (d), 5.5(d))

r m

ě 0, and 1 ě

r1 n

ą

c “ pτ0 , ϕ1 , ϕ2 , τ3 q, n “ pq ´ r, p ` r, rqp“ pr1 , q1 , p1 qq c1 “ pτ0 , ϕ1 , ϕ2 , τ3 q, n1 “ pr1 , q1 , p1 q

Thus pc, nq “ pc1 , n1 q. 246

q1 m`n

ą

247

Bases for Quantum Demazure modules-I

In view of the discussion above we obtain a map θ : L ÝÑ C, which induces a map θ : L ÝÑ C in a obvious way (as above, we identify an element pp1 , q1 , r1 q of L where q1 “ p1 ` r1 , r1 ď n, q1 ď r1 ` m, with the element pr1 , q1 , p1 q). 5.8 We now define a map ψ : C ÝÑ L. Let x P C. Let xmin “ pc, nq. 211 We distinguish the following cases. Let n “ pa, b, cq. ( We follow the convention in 4.4) Case 1. pc, nq is of Type I. This implies 1 ě ma ě bm ` n ě cn ě 0. We have b ě c (since b c m`n ě n ) We set # pc, b, aq ψpxq “ pb ´ c, a ` c, cq

if b ě a ` c if b ă a ` c

b a ñ ma ě b´a (note that ma ě m`n m and hence b ě a ` n, since m ě 1. c Also c ě n, since n ě 1. Thus pc, b, aq P L1 , (in the case b ě a ` c) and pb ´ c, a ` c, cq P L2 (in the case b ă a ` c)).

Case 2. pc, nq is of Type II. This implies tht 1 ě pc, a ` b, bq P L1 , and we set

a n

ě

b m

ě

c n

ě 0. Then

b n

ě

ψpxq “ pa, a ` b, bq Case 3. pc, nq is of Type III. This implies that 1 ě Then pc, a ` b, bq P L2 , and we set

a m

ě

c m

ě 0.

ψpxq “ pa, a ` b, bq Case 4. pc, nq is of Type IV. b ě mc ě 0. Then we set (as in case This implies that 1 ě an ě n`m 1) # pc, b, aq, if b ě a ` c ψpxq “ pb ´ c, c ` a, cq, if b ă a ` c 247

248

V. Lakshmibai

(note that pc, b, aq P L2 (if b ě a ` c) and (b ´ c, c ` a, cq P L1 (if b ă a ` c)). It is easily checked that ψoθ “ IdL and θoψ “ Idc . Thus we obtain Theorem 5.9. The map θ : L Ñ C is a bijection

6 An A-basis for VA 212

6.1 For r P bbz, s P Z` , we set „ rrsrr ´ 1s ¨ ¨ ¨ rr ` 1 ´ ss r “ S rss ¨ ¨ ¨ r1s Where for m P Z, rms “ Let

νm ,´ν´m . ν´ν´1

Let α “ ai , a1 “ α j , where ai j “ ´1.

Fα`α1 “ νFα Fα1 ´ Fα1 Fα .

(1)

We have ([L1]), pM´1q

pMq

Fα Fα1 “ ν´1 Fα`α1 Fα

pMq

` ν´M Fα1 Fα pNq

pNq

Fα`α1 ` ν´N Fα1 Fα Fα Fα Fα1 “ v´1 FαN´1 1

(2) (3)

Hence we obtain (using (1)) pN´1q

Fα1

pNq

pNq

Fα Fα1 “ rN ´ 1sFα1 Fα ` Fα Fα1

pM´1q

Fα Fα1 Fα1

pMq

“ rM ´ 1sFα1 Fα

pMq

` Fα Fα1

(4) (5)

Lemma 6.2. Let α, α1 be as above. For t, u, v P N, we have, ptq puq pvq Fα Fα1 Fα

ˆ ˙ t ÿ t ` v ´ u pu´pt´ jqq pv`tq pt´ jq “ Fα Fα Fα1 j j“t´k

where k “ minpu, tq. This is proved by induction on t using (2)-(5) above. 248

249

Bases for Quantum Demazure modules-I

6.3 Let Bd be the canonical A-basis for VA as constructed in [L2]. Then we have ([L3]), Bd “ B1 Y B2 , where pcq

pbq

paq

1

pb1 q

pa1 q

pb1 q

pa1 q

B1 “ tF1 F2 F1 e, pa, b, cq P L1 u

B2 “ tFcc F1 F2 e, pa1 , b1 , c1 q P L2 u pc1 q

pcq

pbq

paq

(here we identify F2 F1 F2 e, with F2 F1 F2 e if pa, b, cq „ pa1 , b1 , pc1 q

pb1 q

pa1 q

c1 q). Note that if pa1 , b1 , c1 q P L2 , and b1 “ c1 ` a1 , then F2 F1 F2 pa1 q pb1 q pc1 q F1 F2 F1 .

“

6.4 Let x P C, xmin “ pc, nq, and n “ a, b, c. We have (4.8) $ pcq pbq paq ’ F1 F2 F1 e, if pc, nq if of Type I ’ ’ ’ &F pcq F pa`bq F pbq e, if pc, nq if of Type II 1 2 1 vx “ pcq pa`bq pbq ’ F F F ’ 2 1 2 e, if pc, nq if of Type III ’ ’ % pcq pbq paq F2 F1 F2 e, if pc, nq if of Type IV

6.5 Let us take an indexing I of L such that

(1) If pp, q, rq, pa, q, bq are in L1 with a ą p, then pp, q, rq preceeds pa, q, bq. (2) if pp1 , q1 , r1 q, pa1 , q1 , b1 q are in L2 , with a1 ą p1 , then pp1 , q1 , r1 q preceeds pa1 , q1 , b1 q. Then via the bijection ψ : C ÝÑ L, we obtain an indexing J of C induced by I. Let M be the matrix expressing the elements in Bd as A-linear combinations of the elements in Bd , for the indexing J of Bd (resp. I of Bd ) Theorem 6.6. M is upper triangular with diagonal entries equal to 1. Proof. Let x P C, xmin “ pc, nq, n “ pa, b, cq. We may suppose that c ‰ 0; for if c “ 0, then v x P Bd obviously. If pc, nq is of Type II or III, then v x P Bd clearly. We now distinguish the following two cases: 249

213

250

V. Lakshmibai

Case 1. pc, nq is of Type I. pcq pbq paq We have v x “ F1 F2 F1 e. Hence, if b ě a ` c, then v x P Bd . Let then b ă a ` c. We have (Lemma 6.2, with α “ α1 , α1 “ α2 , t “ c, u “ b, v “ a), c „ ÿ c ` a ´ b pb´pc´ jqq pa`cq pc´ jq vx “ F2 F1 F2 e (˚) j j“0 b ě nc ě 0. (Note that pc, nq being of Type I, we have 1 ě ma ě m`n Hence b ą c, and min pc, bq “ c). Now on R.H.S of (˚), each term pb´lq pa`cq plq F2 F1 F2 e is in L2 , since l ď c ď n, a ` c ď m ` c (as a ě m), and a ` c ą b ´ l ` lp“ bq. pc, nq in C corresponds to the element pb ´ c, a ` c, cq in L (under the indexing J (resp. I) for C (resp. L)). Also, it is clear that all the other terms (on the R. H.S. of (˚)) succeed pb´cq pa`cq pcq F2 F1 F2 ( in the indexing I for Bd ).

214

Case 2. pc, n) is of Type IV. The discussion is exactly similar to that of case 1. Theorem 6.7. Bd is an A-basis for VA . Proof. This follows fro Theorem 6.6, since Bd is an A-basis for vA .

7 Basis for quantum Demazure modules 7.1 Let x P C, xmin “ pc, nq, n “ pa1 , a2 , a3 ). Let τ x be as in 3.14. Then τ x is given as follows. If a1 “ a2 “ a3 “ 0, then τ x “ Id. Let r be the largest integer ď 3 such that ar ‰ 0. (1) r “ 1. This implies that c is of Type I or IV. We have τ x “ τ1 presp.ϕ1 ) if c is of Type I (resp. IV). (2) r “ 2. This implies that τ x “ τ2 , if c is of Type I or II and τ x “ ϕ2 , if c is of Type III or IV (3) r “ 3. This implies that τ x “ τ3 . 250

251

Bases for Quantum Demazure modules-I

Theorem 7.2. Let w P W. Let Bw “ tv x : τ x ď wu. Then Bw be an A-basis for Vw,A . Proof. Let Xpϕq be a moving divisor in Xpwq, moved, by α. Then we see easily that ´ Vw,A “ Uα,A Vϕ,A (˚) ´ where Uα,A is the A-submodule of U generated by Fαr , 4 P Z` . For w “ τ0 p“ Idq, the result is clear. For w “ τ3 , the result follows from Theorem 6.7. prq

(1) Let w “ τ1 . Then (˚) implies that tF1 e, r P Z` u generates Vw,A . prq

prq

Now F1 e “ 0, for f ą m. Hence tF1 e, 0 ď r ď mu is an a prq

A-basis for Vw,A , while Bw is precisely tF1 e, 0 ď r ď mu

(2) Let w “ ϕ1 . The proof is similar as in (1). pqq

(3) Let w “ τ2 and ϕ “ τ1 . Then we have (in view of (˚)), tF2 v, v P 215 pqq

prq

Bϕ u generates Vw,A . We have F2 F1 e “ 0, if q ą r ` n (4.9). pqq

prq

Hence tF2 F1 e, r ď m, q ď r`nu generates Vw,A as an A-module. Now, if

r m

ě

q m`n ,

pqq

prq

then F2 F1 e “ v x , where x “ c, n, c “

pτ0 , τ1 , τ2 q, n “ pr, qq; if

q m`n

ą

r m,

pqq

prq

then F2 F1 e “ v x , where

x “ pc, nq, c “ pτ0 , ϕ1 , ϕ2 q, n “ pq ´ r, rq (6.4). Hence we see that Bw generates Vw,A . The linear independence of Bw follows Theorem 6.7 (since Bw Ď Bd ). (4) Let w “ ϕ2 . The proof is similar to that in (215).

8 Appendix We have used the results of [L2] mainly to prove that #C “ dimVd , (§5). We can get around proving #C “ dimpVd q, by showing that #C “ #t standard Young tableaux on S Lp3q of type pm, nqu. We can then prove 251

252

V. Lakshmibai

the results of §6, §7 in the same spirit as in [LS]. In this Appendix, we establish a bijection between C and tstandard Young tableaux on S Lp3q of type pm, nqu. 8.1 Let G “ S Lp3q. Let P1 “ tId, s1 u, P2 “ tId, s2 u. Let us denote the set of minimal representatives of WP1 (resp. WP2 ) in W by Θ “ tΘ1 , Θ2 , Θ3 u (resp. Λ “ tλ1 , λ2 , λ3 u). Then Θ (resp. Λ) is totally ordered (under the Bruhat order ľ ). Let θ3 ľ θ2 ľ θ1 ; λ3 ľ λ2 ľ λ1 . Let X “ tθ3 , θ2 , θ1 , λ3 , λ2 , λ1 u We have a partial order ě on X given as follows. Lets x, y P X. The x ě y, if either both x, y P Θ (resp. Λ), and x ľ y, or x (resp. y) P Θpresp.Λq, and px, yq ‰ pθ1 , λ3 ). A classical standard Young tableau on G of type pm, nq can be noted as τ1 τ12 ¨ ¨ ¨ τ1m τ21 τ22 ¨ ¨ ¨ τ2n where τ1 j (resp. τ2k ) P Θ (resp. Λ ) and τ11 ě τ12 ě ¨ ¨ ¨ ě τ1m ě τ21 ě τ22 ě ¨ ¨ ¨ ě τ2n Let Y “ tstandard Young tableaux of type pm, nqu 216

8.2 Let a P Y, say a “ τ11 ¨ ¨ ¨ τ1m τ21 ¨ ¨ ¨ τ2n . We define the integers ra , qa , pa , ua , ta , sa as follows. ra “ #tτ1 j : τ1 j “ θ3 u

pa “ #tτ1 j , τ2k : τ1 j “ θ2 , τ2k “ λ3 u

qa ´ ra “ #tτ2k : τ2k “ λ3 or λ2 u ua “ ra

ta “ ra ` pa

sa “ qa ´ ra

Note that ra ě m, qa ď ra ` n, ra ` pa ď qa ´ ra ` m. Note also that ra , pa , qa (resp. ua , ta , sa ) completely determine “a”. 252

253

Bases for Quantum Demazure modules-I

8.3 The map f : Y Ñ C. For a P Y, we define f paq “ pca , na q as follows. For simplicity of notation let us drop off the suffix ‘a1 is in ra , ¨ ¨ ¨ , sa , ca , na . (We follow the convention in 4.4) while denoting a chain pc, nq). We first observe that r q q ´ r p ` 2r ´ q p ` r s t ´ s u ` s t , , , , , , , , , m m`n n m m`n n m`n m`n m`n are all ď 1. We now distinguish the following cases. q Case 1. 1 ě mr ě m`n ě np ě 0 We set c “ pτ0 , τ1 , τ2 , τ3 q, n “ pr, q, pq q r Case 2. m`n m q r ą mr ðñ q´r Now m`n n ą m We divide this case into the following cases. p`r Case 2(a). 1 ě q´r n ą m`n ě f racrm ě 0. This is equivalent to

1ě

s t u ě ě ě0 n m`n m

We set c “ pτ0 , ϕ1 , ϕ2 , ϕ3 q, n “ ps, t, uq.

p`r ě f racq ´ rn ě Case 2(b). 1 ě m`n This is equivalent to

1ě

r m

ě0

q´r r p ` 2r ´ q ě ě ě0 m n m

We set c “ pτ0 , ϕ1 , τ2 , τ3 q, n “ pp ` 2r ´ q, q ´ r, rq

r Case 2(c). 1 ě q´r n ě m ě This is equivalent to

1ě

p`r m`n

ě0

r p q´r ě ě ě0 n m n

We set c “ pτ0 , τ1 , ϕ2 , τ3 q, n “ pq ´ r, r, pq 253

217

254

V. Lakshmibai

q q ď mr , m`n ă np Case 3. m`n q q q , ě mr ðñ q´r Now, m`n n ă m`n , and m`n ă q q´r q p m`n ă n Hence in this case, we have n ď m`n ă that q ă r ` p, i.e., u ` s ă t. We divide this case into the following subcases.

p n p n.

ðñ q´p m ă This implies

t ě mu ě 0 Case 3(a). 1 ě ns ě m`n We set c “ pτ0 , ϕ1 , ϕ2 , τ3 q, n “ ps, t, uq

t t s Case 3(b). m`n ě ns . Now m`n ě ns ðñ t´s m ě n . The condition u`s t´s u`t s ă t´s u ` s ă t implies m`n m (for, otherwise), m`n ě m ě n ñ u`s´pt´sq n

ě

t´s m

ě

s n

ñ u ` s ě t, which is not true) Hence, either.

t´s u`s s ě ě ě 0(or) m m`n n t´s s u`s 1ě ě ě ě0 m n m`n Now (1) is equivalent to 1ě

1ě

(1) (2)

q q´r p ` 2r ´ q ě ě ě0 m m`n n

and we set c “ pτ0 , τ1 , τ2 , τ3 q, n “ pp ` 2r ´ q, q, q ´ rq, if (1) holds. Similarly, (2) is equivalent to 1ě

s u t´s ě ě ě0 m n m

and we set c “ pτ0 , τ1 , ϕ3 , τ3 q, n “ pt ´ s, s, uq, if (2) holds. 218

8.4 The map g : C ÝÑ Y. Let x P C, xmin “ pc, nq, n “ pa, b, cq. Set gpxq “ a, where rp“ ra q, qp“ qa q, pp“ pa q are given as follows. # pa, b, cq pr, q, pq “ pb, a ` b, cq 254

if pc, nq is of Type I if pc, nq is of Type II

Bases for Quantum Demazure modules-I # pb, a ` b, cq, if pc, nq is of Type III ps, t, uq “ pa, b, cq, if pc, nq is of Type IV

255

(where, recall that u “ r, t “ r ` p, s “ q ´ r). It is easily checked that go f “ IdY , f o f “ IdC . Thus we obtain Theorem 5. The map f : Y ÝÑ C is a bijection.

References [B] N. Bourbaki, Groupes et algebres de Lie simisimples, Chaptires IV, V et VI., Bourbaki (1961/62). [B-W] A. Bj¨orner and M. Waches, Bruhat order of Coxeter groups and shellability, Adv. Math. 43 (1982) 87–100. [D] V. G. Drinfeld,Hopf algebras and the Yang-Baxter equation, Soviet Math. Dokl. 32 (1985) 254–258. [J]

M. Jimbo, A q-difference analioure of Upgq and the Yang-Baxter equation, Lett. Math. Phsy. 10 (1985) 63–69.

[L1] G. Luszting, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, Jour. Amer. Math. Soc. 3 (1990) 257–296. [L2] G. Luszting, Canonical baases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990) 447–498; II, progress of Theor. Physics (to appear). [L3] G. Luszting, Introduction to quantized enveloping algebras (preprint). [L4] G. Lusztig, Problems on Canonical Bases (preprint). [LS] V. Lakshmibai and C.S. Seshadri, Geometry of G{P ´ V, J. Alge- 219 bra, 100 (1986) 462–557. 255

256

V. Lakshmibai

Northeastern University Boston, MA 02115 U.S.A.

256

An Appendix to Bases for Quantum Demazure modules-I Let g be symeetrizable Kac-Moody Lie algebra, and U be the quantized 221 enveloping of g as constructed by Drinfeld (cf [D]) and Jimbo (cf [J]). This is an algebra over Qpvq (v being a parameter) which specializes ti Upgq for v “ 1, Upgq being the universal enveloping algebra of g. This algebra has agenerators Ei , Fi , ki , 1 ď i ď n, which satisfy “the quantum Chevalley and Serre relations”. Let U ˘ be the Qpvq-sub algebra of U generated by Ei (respFi ), 1 ď i ď n. Let A “ Zrv, v´1 s and U A˘ be the prq

A-subalgebra of U generated by Ei prq

prq

(resp. Fir ), 1 ď i ď n, r P Z` ,

(here Ei , Fi are the quantum divided powers of (cf [J])). Let λ be a dominant, integral weight and Vλ the associated simple U-module. Let us fix a highest weight vector e in Vλ and denote VA “ U Ae p“ U A´ eq. Let W be the Weyl group of g. For w P W, let ew be the corresponding extremal weight vector in Vλ of wight wpλq. Let Vw “ U ` ew , Vw,A “ U A` ew . In [La] (see also [LS]), we proposed a conjecture (which we recall below) towards the construction of an A-basis for VA compatible with tVw,A , w P Wu. This conjecture consists of two parts. The first part givews a (conjectural) character formula for the U ` -module Vw in terms of certain weighted chains in W. The second part gives a conjectural A-basis Bλ for VA , compatible with tVw,A w P Wu. We now state the conjecture.

Part I Iλ, an indexing set for Bλ Let λ “ Σdi ωi , ωi being the fundamental weights. Admissble weighted λ-chains Let c “ tµ0 , µ1 , . . . µr u be a λ-chain in W, i.e., µi µi´1 , ℓpµi q “ ℓpµi´1 q` 1 (if dt “ 0 for t “ i1 , . . . , i s , then we shall work with wQ , the set of minimal representatives of WQ in W, WQ being the weyl group of the 257

258 parabolic subgroup Q, where S Q “ ptαt , t “ i1 , ¨ ¨ ¨ , i s uq). 222 Let µi´1 “ si µi , where βi is some positive real root. Let pµi´1 pλq, β˚ q “ mi . A 1. Definition. A λ -chain c is called simple if all β1i s are simple. A 2. Definition. m By a weighted λ chain we shall mean pc, nq where c “ tµ0 , ¨ ¨ ¨ , µr u is chain and n “ tn1 , . . . , nr u, ni P Z` . A 3. Definition.A weighted λ-chain pc, nq is said to be admissible if 1 ě mn11 ě ¨ ¨ ¨ ě mnrr ě 0. ! Let pc, nq be admissible. Let us denote the unequal values in mn12 , ¨ ¨ ¨ ) nr mr by a1 , ¨ ¨ ¨ , a s so that 1 ě a1 ą a2 ą ¨ ¨ ¨ ą a s ě 0. Let i0 ¨ ¨ ¨ , i s be defined by nj “ at , it´1 ` 1 ď j ď it . i0 “ 0, i s “ r, mj We set Dc,n “ tpa1 , . . . , a s q; pµi0 , . . . , µis qu A 4. Definition. Let pc, nq, pc1 , n1 q be two admissible weighted λ-chains. Let Dc,n “ tpa1 , . . . , a s q; pµi0 ¨ ¨ ¨ , µis qu, and Dc1 ,n1 “ tpa11 , ¨ ¨ ¨ , a1l q; pτ j0 , . . . , τ jl qu. We say pc, nq „ pc1 , n1 q, if s “ t, and at “ a1t , it “ jt , µit “ τ jt ,1 ď t ď s. Let Cλ “ t all admissible weighted λ-chainsu, and Iλ “ Cλ { „. Let π P Iλ , and let pc, nq be as representative of π. With notations as above, we set s ÿ τpπq “ µis , vpπq “ pat ´ at`1 qµit pλq t“0

where a0 “ 1 and a s`1 “ 0 (note that τpπq and vpπq depend only on π and not on the representative chosen). For w P W, let Iλ pwq “ tπ P Iλ | w ě τpπqu.

A 5. Conjecture. char Vw “

ÿ

πPIλ pwq

258

evpπq

An Appendix to Bases for Quantum Demazure modules-I

259

Part II: An A-basis for V A compatible with tVw, A , w P Wu Let π, pc, nq etc. be as in Part I. To pc, nq there corressponds a canonical (not necessarily admissible) weighted chain pδpcq, δpnqq with δpcq simple (cf [La],3.8). Let δpcq “ tθ “ τo , . . . , τr u, n “ tn1 , ¨ ¨ ¨ , nr u, βt “ αit , 1 ě t ě r (note that βt ’s are simple). We set pn q

pn q

vc,n “ Fir r ¨ ¨ ¨ Fi1 1 eθ A 6. Conjecture. For each π P Iλ , choose a representative pc, nq for π. 223 Then tvc,n : w ě τpπqu is A-basis for Vw,A . In [Li], Littelmann proves Conjecture 1, and as a consequence gives a Littlewood-Richardshon type “decomposition rule” for a symmetrizable KacMoody lie algebras g, and a “ restriction rule” for a Levi subalgebra L of g which we state below. Let θ be a dominant integral weight and let π P Iθ . Let pc, nq be a representative of π and let Dc,n be as above. Let us denote ppπ, θq “

#

s ÿ

k“t

pak ´ ak`1 qµik pθq, 0 ď t ď s

+

A 7. Definition. Let λ, θ be two dominant, integral weights. Let π P Iθ . Then π is said to be λ-dominant if λ ` ppπ, θq is contained in the dominant Weyl chamber. A 8. Definition. Let L be a Levi subalgebra of g, and let π P Iλ . Then π is said to be L-dominant if ppπ, λq is contained in the dominant Weyl chamber of L. Decomposition rule. ([Li]) Let λ, µ be two dominant integral weights. Let Ipλ, µq “ tπ P Iµ | π is λ ´ dominantu. Then Vλ b Vµ “

â

πPIpλ,µq

259

Vλ`vpπq

260 Restriction rule. ([Li]) Let L be a Levi subalgebra of g . Let Ipλ, Lq “ tπ P Iλ : π is L-dominantu. Then à resL Vλ “ Uvpπq πPIpλ,Lq

(here, for an integral weight θ contained in the dominant Weyl chamber of L, Uθ denotes the corresponding simple highest weight module of L) In [Li], Littelmann introduces operators eα , fα on Iλ , (for α simple), and associates an oriented, colored (by the simples roots) graph GpVλ q α with Iλ as the set of vertices, and π Ý Ñ π1 if π1 “ fα pπq. He conjectures that GpVλ q is the crystal graph of Vλ as constructed by Kashiwara ([K]). Using the decomposition rule, Littlemann gives in [Li] a new (and simple) proof of the Parthasarathy-Ranga Rao Varadarajan conjecture.

References 224

[D] V.G. Drinfeld, Hopf algebras and the Yang-Baxter equation, Soviet Math. Dokl. 32 (1985) 254-258. [J]

M. Jimbo, A q-difference analogue of Upgq and the Yang-Baster equation, Lett. Math. Phys. 10 (1985) 63-69.

[K] M. Kashiwara, Crystalizing the Q-analogue of Universal Enveloping Algebras, Commun. Math. Phys. 133 (1990) 249-260. [La] V. Lakshmibai, Bases for Quantum Demazure modules, (The same volume). [Li] P. Litttelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras(preprint). [LS] V. Lakshmibai and C.S Seshadri, Standard Monomial Theory, “Proceedings of Hyderbad conference on Algebraic Groups”, Manoj Prakashan, Madras (1991) 279-323.

260

An Appendix to Bases for Quantum Demazure modules-I Northeastern University Mathematics Department Boston, MA 02115, U.S.A.

261

261

Moduli Spaces of Abelian Surfaces with Isogeny* Ch. Birkenhake and H. Lange To M.S. Narasimhan and C.S Seshadri on the occasion of their 60th birthdays 225

Let pX, Lq be as polarized abelian surface or type p1, nq. An isogeny of type p1, nq is an isogeny of polarized abelian surfaces π : pX, Lq Ñ pY, Pq such that P defines a principle polarization on Y. According to [H-W] the coarse moduli space A1,n of such triplets pX, L, πq exists and is analytically isomorphic to the quotient of the Siegel upper half space of degree 2 by the action of Γ “ tM P S p4 pZq : M “ pmi j q with n|mi4 , i “ 1, 2, 3u. Ap1,nq is a finite covering of themodulinpsace of principally ploarixed abelina surface as well as of the moduli space of polarized abelian surface of type p1, nq. On the other hand, the moduli space of polarized abelian surfaces with level n-structure is as finite covering of Ap1,nq . If for example n is a prime, the degrees of these coverings are pn ` 1qpn2 ` 1q, pn ` 1q and npn ´ 1q respectively The aim of the present papert is to give explicit algebraic descriptions of the moduli spaces A1,2 (see Theorem 3.1) and A1,3 (see Theorem 6.1). An immediate consequence is that the moduli spaces A1,2 and A1,3 are rational. An essential ingredient of the proof is the fact that the moduli space A1,n is canonically isomorphic to the moduli space Cn2 of cyclic e´ tale n-fold coverings of curves of genus 2. This will be shown in Section 1. The second important tool is the fact that the composition of every C Ñ H in Cn2 with the hyperlliptic covering H Ñ P1 is Galois with the dihedral group Dn as Galois group (see Section 2). Finally we need some results on duality of polarizations on abelian surfaces which we compile in Section 4. We would like to thank W. Barth and W.D. Geyer for some valuable discussions. * Supported

by DFG-contract La 318/4 and EC-contract SC1-0398-C(A)

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263

1 Abelian Surface with an Isogeny of Type p1, nq 226

In this section we show that there is a canonical isomorphism between the moduli space of polarized abelian surfaces with isogeny of type p1, nq and the moduli space of cyclic e´ tale n-fold coverings of curves of genus two. Let X be an abelian surface over the field of complex numbers. Any ample line bundle L on X defines a polarization on X. In the notation we do not distinguish between the line bundle L and the corresponding p “ Pic0 pXq the dual abelian variety. The polarization. Denote by X polarization L determines an isogeny p φL : x Ñ X,

x ÞÑ t˚x L b L´1

where t x : X Ñ X is the translation map y map y` x. The kernel KpLq of ϕL is isomorphic to pZ{n1 Z ˆ Z{n2 Zq2 for some positive integers n1 , n2 with n1 | n2 . We call pn1 , n2 q the type of the polarization. Any polarization of type pn1 , n2 q is the n1 -th power of a unique polarizations of type p1, nn21 q. Hence for moduli problems it suffices to consider polarizations of type p1, nq. From now on let L be a line bundle defining a polarization of type p1, nq. An isogeny of type p1, nq is by definition an isogeny of polarized abelian varieties π : pX, Lq Ñ pY, Pq whose kernel is cyclic of order n. Necessarily Pm defines a principal polarization on Y and ker p is contained in KpLq. Conversely, according to [L-B] Cor. 6.3.5 any cyclic subgroup of KpLq of order n defines an isogeny of type p1, nq of pX, Lq. In particular, if n is a prime number, then pX, Lq admits exactly n`1 isogenies of type p1, nq. According to [L-B] Exercise 8.4 the moduli space A1,n of polarzed abelian sufaces with isgeny of type p1, nq exists and is analytically isomorphic to the quotient of the Siegel upper half space h2 of degree 2 by the group tM P S p4 pZq|M “ pmi j q with n|mi4 , i “ 1, 2, u. In the sequel a curve of genus two means either a smooth projective curve of genus 2 or a union of two elliptic curves intersecting transversally at the origin. Note that such a union E1 ` E2 is of arithmetic genus 2. Torelli’s Theorem implies that the moduli space of principally polar263

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ized abelian surfaces can be considered as a moduli space for curves of genus two in this sense. Let f : C Ñ H be a cyclic e´ tale covering of degree n of a curve H of genus 2. According to Hurwitz’formula formular C has arithmetic genus n ` 1 . Every line bundle l P Pic0 pHq of order n determines such a cyclic e´ tale covering f : C Ñ H (for an explicit description of 227 the covering see Section 2). Two such line bundles lead to the same covering, if they generate the same group in Pic0 pHq. This implies that the (coarse) moduli space Cn2 of cyclic e´ tale n-fold coverings of curves of genus two is a finite covering of the moduli space M2 of curves of genus two. In particular Cn2 is an algebraic variety of dimension 3. The moduli spaces A1,n and Cn2 are related as follows. 1.1 Propostion There is a canonical biholomorphic map Ap1,nq Ñ Cn2

There seems to be no explicit construction of the moduli space Cn2 in the literature. One could also interprete Proposition 1.1 as a construction of Cn2 . However it is not difficult to show its existence in a different way and thus the proposition makes sense as stated. Proof. Step I: The map A1,n Ñ Cn2 . Let π : pX, Lq Ñ pY, Pq be an isogeny of type p1, nq. We may assume that π˚ P » L as line bundles. Since pY, Pq is a principally polarized abelian surface there is curve H of genus 2 (in above sense) such that Y “ JpHq, the Jacobian pf H, and P » OY pHq. Note that for H “ E1 ` E2 with elliptic curves E1 and E2 , JpHq “ Pic0 pHq » E1 ˆ E2 . By assumption C : π´1 H P |L|. The e´ tale covering π : X Ñ Y is given by a line bundle l P Pic0 pYq of order n and the coverin π|C : C Ñ H corresponds to l|H. Since the restriction map „ Pic0 pYq Ý Ñ Pic0 pHq is an isomorphism, the line bundle l|H is of order n and thus π|C : C Ñ H is an element of Cn2 . Step II: The inverse map A1,n Ñ Cn2 . Let f : C Ñ H be a cyclic e´ tale covering in Cn2 associated to the line bundle lH P Pic0 pHq. Via the „ isomorphism Pic0 pJpHqq Ý Ñ Pic0 pHq the line bundle lH extends to a line bundle l P Pic0 pJpHqq of order n. Let π : X Ñ Y “ JpHq denote the cyclic e´ ta;e n-fold covering associated to l. Then L “ π˚ OY pHq defines a polarization of type p1, nq, since KpLq is a finite group of order 264

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n2 (by Riemann-Roch) and contains the cyclic group ker π of order n. Hence π : pX, Lq Ñ pY, OY pHqq is an element of Ap1,nq . Obviously the maps Ap1,nq Ñ Cn2 and Cn2 Ñ Ap1,nq are inverse to each other. Finally, extending the above construction to families of morphisms of curves and abelian varieties one easily sees that the maps are holomorphic.

´ 2 Cycli Etale Coverings of Hyperelliptic Curves Any curve H of genus 2 (in the sense of section 1) admits a natural involution ι with quotient H{ι of arithmetic genus 0. The aim of this section is to show that for any finite cyclic e´ tale covering f : C Ñ H the 228 composition C Ñ H Ñ H{ι is Galois and to compute its Galois group. We prove the result in greated generality than actually needed, since this makes no difference for the proof. In this section a hyperelliptic curve means a complete, reduced, connected curve admitting an involution whose quotient is of arithmetic genus zero. Let H denote a hyperelliptic curve of arithmetic genus g over k with hyperelliptic covering H Ñ P. Suppose f : C Ñ H is a cyclic e´ tale covering of degree n ě 2. We first show that the composed map C Ñ P is a Galois covering with the dihedral group Dn of order 2n as Galois group. Let ι : H Ñ H denote the hyperelliptic involution and τ : C Ñ C an automorphism generating the group GalpC|Hq. There is a line bundle L P Pic0 pHq with Ln » OH such that C “ SpecpAq with A :“ OH ‘ L ‘ ¨ ¨ ¨ ‘ Ln´1 and where the OH -algebra structure of A is given by an „ isomorphism σ : OH Ý Ñ Ln . Consider the pull back diagram i

SpecpĂ˚ Aq “Ă˚ C

/ C “ Spec A f

H

Ă

/H

Since ι˚ L “ L´1 , the isomorphism σ induces isomorphisms σν “ pσ b 1L´ν q ˝ ι˚ : Lν Ñ Ln´ν 265

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for ν “ 0, . . . , n ´ 1 which yields an OH -algebra isomorphism A Ñ ι˚ A. Hence we may identify ι˚C “ C and j : C Ñ C is an automorphism. 2.1 Proposition. The covering C Ñ P is Galois with GalpC|P1 q “ Dn Proof. If suffices to show jτ j “ τ´1 . Accordings to [[EGAI], Th.9.1.4] the automorphism τ of C “ Spec A corresponds to an OH -algebra automorphism τ˜ : A Ñ A, namely ˆ ˙ 2πi n´1 τ˜ pa0 , a1 , . . . , an´1 q “ pa0 , ξa1 , . . . , ξ an´1 q with ξ “ exp . n Similarly using [[EGAI], Cor. 9.1.9] the automorphism j of C corresponds to the algebra automorphism ˜ : A Ñ A over ι˚ defined by ˜pa0 , a1 , . . . , an´1 q “ pσ0 pa0 q, σn´1 pan´1 q, . . . , σ1 pa1 qq. We have to show that ˜τ˜ , ˜ “ τ˜ ´1 . But σn´ν ξn´ν σν paν q “ ξ´ν aν . This implies the assertion. 229

The dihedral group Dn contains the involutions jτν for ν “ 0, . . . , n ´ 1 n and for even n also τ n . These involutions correspond to double covern ings C Ñ Cν “ C{ jτν for ν “ 0, . . . , n ´ 1 and C Ñ C 1 “ C{τ 2 for even n. If C is smooth and irreducible we have for the genera gc1ν and gc1 of Cν and C 1 2.2 Proposition. a) For on odd: gcν “ 12 pg ´ 1qpn ´ 1q for ν “ 0, . . . , n ´ 1. b) For n even: p n2 ´ 1qpg ´ 1q ď gν ď 12 npg ´ 1q for all 0 ď ν ď n ´ 1 and gc1 “ n2 ` 1. The proof is an application of the formula of Checvalley-Weil (see [C-W]). We omit the details. The genus of C 1 can be computed by Hurwitz’ formula since C Ñ C 1 is e´ tale. 2.3 Remark. Let H “ E1 ` E2 be a reducible curve of genus two as in Section 1. The curve H is hyperelliptic with hyperelliptic involution 266

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267

ι the multiplication by -1 on the each curve Ei . The quotient P “ H{ι consists of two copies of P1 intersecting in one point. In this situation Proposition 2.1 can be seen also in the following way. If for example the covering Ñ H is nontrivial on each component Ei , then C consists of two elliptic curves F1 and F2 intersecting in n points. We choose one of these points to be the origin of F1 and F2 the remaining intersection points are x. . . . , pn ´ 1qx for some n-division point x on F1 and F2 . The automorphism τ : C Ñ C defined as the translation t x by x on each Fi generates the group of covering transformations of C P H. The involution ι on H lifts to an involution j on C, the multiplication by (-1) on each Fi . Obviously jτ j “ τ´1 , so C Ñ H{ι is a Galois covering with Galois group Dn “ą j, τ ą. As in the irreducible case we consider the double coverings C Ñ Cν “ C{ jτν for π ν “ 0, . . . , n ´ 1 and C Ñ C 1 “ C{τ 2 for even n. Also here the result of Proposition 2.2 is valid: for example, if n is odd and the covering C Ñ H is nontrivial on each component, then Cν consists of two copies n´1 of P1 intersection in n`1 2 points, the images of kx for k “ 0, . . . , 2 . In particular Cν has arithmetical genus n´1 2 . The other cases can be worked out in a similar way.

3 The Moduli Space A01,2 Denote by A01,n the open set in A1,n corresponding to abelian surfaces of type p1, nq with and isogeny onto a Jacobian of a smooth curve of genus 2. The aim of this section is to give a description of the moduli 230 space A01,2 . From this it is easy to see that A1,2 is rational. Let A˜ 1 the modulo space of elliptic curves E together with a set of four points of E of the form t˘p1 , ˘p2 u. Necessarily such a set does not cotain any 2-division point of E. We write the elements of A˜ 1 as pairs pE, t˘p1 , ˘p2 uq. The main result of this section is 3.1 Theorem. The moduli space A01,2 of polarized surfaces with an isogeny of type p1, 2q onto a Jacobian of a smooth curve of genus 2 is canonically isomorphic to A˜ 1 . 267

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3.2 Corollary. The moduli space Ap1,2q is rational. Proof of Corollary 3.2 Via the j-invariant, U :“ C ´ t0, 1728u is the moduli space of elliptic curves without nontirivial automorphisms. U admits a universal elliptic curve p : E Ñ U. Consider the quotient p : E{p´1q Ñ U by the action of p´1q on every fibre. And open set of A˜ 1 can be identified with an open set of the relative symmetric product S 2p pE{p´1qq over U. Every fibre of S 2p pE{p´1qq Ñ U is isomorphic to P2 , so S 2p pE{p´1qq is a P2 -bundle over U. According to [G] Corollaire 1.2 the Brauer group of U is zero. Hence S 2p pE{p´1qq is the projectivization of a vector bundle on U and thus it is rational. 0 ˜ For the proof of Theorem 3.1 we first describe the map Ap1,2q Ñ A1 .

Let pX, L, πq be an element in A0p1,2q and f : C Ñ H the corresponding e´ tale double covering of a curve H of genus 2 according ot Proposition 1.1. As we say in the last section the automorphism group of C contains the group D2 . As above denote by τ P D2 the involution corresponding to the covering C Ñ H and j P D2 a lifting of the hyperelliptic involution on H. Either from the proof of Proposition 2.2 or by considering the ramification points of the 4-fold covering C Ñ P1 one easily sees that the genera of the curves C{ j and C{ jτ are 0 and 1. By eventually interchanging the roles of j and jτ we may assume that E “ C{ j is an elliptic curve and C{ jτ “ P1 . In particular the curve C is hyperelliptic and we have a commutative diagram C

/E

/ P1

f

H

We can choose the origin in E in such a way that (-1) is the involution on E corresponding to the covering E Ñ P1 , so that the ramification 231 points of E Ñ P1 are the 2-division points of E. From the commutative diagram we see that the 4 ramification points p1 , . . . , p4 P E of the covering c Ñ E are different from the 2-division points of E. Since the involutions j and τ commute and τ is a lifting of (-1) on E, the involution 268

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269

(-1) acts on the set tp1 , . . . , p4 u. Hence we may assume that p3 “ ´p1 and p4 “ ´p2 . Now define the map ψ : A0p1,2q Ñ A˜ 1 ,

pX, L, πq ÞÑ pE, t˘p1 , ˘p2 uq.

Since E and the set t˘p1 , ˘p2 u can be given via algebraic equations out of the covering C Ñ H, the map ψ is holomorphic and it remains to show that it admits an inverse. Let pE, t˘p1 , ˘p2 uq P A˜ 1 . Note that E admits exactly four double coverings ramified in ˘p1 and ˘p2 , since the line bundle OE pp1 ` p2 ` p´p1 q ` p´p2 qq admits exactly 4 square roots in Pic2 pEq. They can be given as follows: Let E be given by the equation y2 “ xpx´1qpx´aq and choose as usual the origin to be the flex at infinity. Then the nontivial 2division points of E are px, yq “ p0, 0q, p1, 0q, pa, 0q. Write pi “ pxi , yi q for i “ 1, 2 and consider the double coverings Di Ñ P1 , i “ 0, . . . , 3, defined by the equations y20 “ xpx ´ 1qpx ´ aqpx ´ x1 qpx ´ x2 q

y21 “ xpx ´ X1 qpx ´ x2 q

y22 “ px ´ 1qpx ´ x1 qpx ´ x2 q

y23 “ px ´ aqpx ´ x1 qpx ´ x2 q

Finally denote by Ci the curve corresponding to the composition of the function fields of E and Di . Then we have the following commutative diagram / Di Ci / P1

E

According to Abhyhankar’s lemma Ci Ñ E is not ramified over the 2-division points of E, hence C0 , . . . C3 are exactly the four double coverings of E ramified in ˘p1 “ px1 , ˘y1 q and p2 “ px2 , ˘y2 q. Moreover D0 is of genus 2 and D1 , D2 , D3 are genus 1 and Ci |P1 is Galois with GalpCi |P1 q “ D2 “ă ji , τi ą where ji and τi are the involutions corresponding to Ci Ñ E and Ci Ñ Di respectively. The third involution in 269

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GalpCi |P1 q is ji τi . The corresponding curves D1i “ Ci { ji τi are given by the equations respectively z20 “ px ´ x1 qpx ´ x2 q

z21 “ px ´ 1qpx ´ aqpx ´ x1 qpx ´ x2 q

z22 “ xpx ´ aqpx ´ x1 qpx ´ x2 q

z23 “ xpx ´ 1qpx ´ x1 qpx ´ x2 q 232

Hence C0 is the only covering of E ramified in ˘p1 and ˘p2 admitting an e´ tale double covering of a curve of genus 2 in this way. So the data pE, t˘p1 , ˘p2 uq determine uniquely an element of C22 , namely C0 Ñ D0 . Let pX, L, πq denote the corresponding element of A01,2 and define a map ϕ : A˜ 1 Ñ A0p1,2q , pE, t˘p1 , ˘p2 uq ÞÑ pX, L, πq. Obviously ϕ is holomorphic and inverse to ψ. This completes the proof of Theorem 3.1. The above proof easily gives another description of the moduli space 0 A1,2 . Let H3 pD2 q denote the moduli space of isomorphism classes of curves of geneus three given by the following equation y2 “ px2 ´ 1qpx2 ´ αqpx2 ´ βqpx2 ´ γq

(1)

with pairwise different α, β, γ P C ˚ ´ t1u. Every curve in H3 pD2 q is hypereliptic and its automorphism group contains D2 “ tx ÞÑ ˘x, y ÞÑ ˘yu which explains the notation. 3.3 Proposition There is a canonical isomorphism A0p1,2q » H3 pD2 q. Proof. Let px, L, πq P A01,2 and C Ñ H be the associated e´ tale double covering in C22 . As we saw in the proof above the curve C is hyperelliptic. Moreover AutpCq contains D2 according to Proposition 2.1. It is well known (see e.g. [I]) that every hyperelliptic curve C of genus three with D2 Ă AutpCq admits an equation of the form (1). Hence the assignment pX, L, pq ÞÑ C gives a holomorphic map A01,2 Ñ H3 pD2 q. 270

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271

For the inverse map suppose C P H3 pD2 q is given by an equation (1). The involution px, yq ÞÑ p´x, ´yq induces the double covering C Ñ H, where H is given by the equation v2 “ upu´1qpu´αqpu´βqpu´γq. It is easy to see that C Ñ H is an element of C22 and the assignment C ÞÑ tC Ñ Hu defines a holomorphic map H3 pD2 q Ñ C22 » A0p1,2q which in inverse to the map A0p1,2q Ñ H3 pD2 q given above.

3.4 Remark. Let U “ tpα, β, γq P pC ˚ ´t1uq3 : α ‰ β ‰ γ ‰ αu. The moduli space H3 pD2 q is birational to the quotient of U by a (nonlinear) action of the group Z2 ˆ S4 . As a consequence of Corollary 3.2 the quotient U{Z2 ˆS4 is rational, which seems not be known from Invariant Theory.

4 Remarks on Duality on Polarized Abelian Surfaces 233

In this section we introduce the dual of a ploarization of an abelian surface and compile some of its properties needed in the next section. The results easily generalize to abelian varieties of arbitrary dimension. Let pX, Lq be a polarized abelian surface of type p1, dq. Recall that the polarization L induces an isogeny from X onto its dual ϕL : x Ñ p ÞÑ t˚x L b L´1 . Its kernel KpLq is isomorphic to the group Z{dZ ˆ X,x Z{dZ.

4.1 Proposition. There is a unique polarization Lˆ on Xˆ characterized by the following two equivalent properties: iq ϕ˚L Lˆ ” Ld

and

iiq ϕLˆ ϕL “ d ¨ 1X

The polarization Lˆ is also of type p1, dq. Proof. The equivalence i) ðñ ii) follows from the equation ϕϕ˚ Lˆ “ L ϕˆ L ϕLˆ ϕL , since the polarization L and the isogeny ϕL determin each other and ϕˆL “ ϕL (see [L-B] Section 2.4). The uniqueness of Lˆ follows from ii) and again, since Lˆ and ϕLˆ determine each other. 271

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ˆ For the existence of Lˆ note that ϕ´1 L exists in Homp X, Xq b Q since ´1 ϕL is an isogeny. By [L-B] Proposition 1.2.6 ψ “ dϕL : Xˆ Ñ X is an isogeny. We have ˆ L ψ “ ψd ˆ “ dψ. ϕψ˚ L “ ψϕ

(1)

ˆ According to [L-B] Lemma 2.5.6 there exists a polarization Lˆ P PicpXq d ˚ ˆ such that L ” φ L and hence ϕψ˚ L “ ϕLˆ d “ dϕLˆ . Together with (1) this implies ψ “ ϕLˆ and thus ϕLˆ ϕL “ d ¨1X . Moreover ii) implies that Lˆ is of type p1, dq. In the next section we need the following example of a pair of dual polarizations. 4.2 Example. Let E be an elliptic curve and Ξ the polarization on E ˆ E defined by the divisor E ˆ t0u ` t0u ˆ E ` ∆, where ∆ denotes 234 the diagonal in E ˆ E. If we identity as usual E “ Eˆ via ϕOE p0q , then we have for the dual polarization Ξˆ on E ˆ E p “ OEˆE pE ˆ t0u ` t0u ˆ E ` Aq, Ξ

where A denotes the antidiagonal in E ˆE. To see this note that Ξ can be written as Ξ “ p˚1 OE p0q ` p˚2 OE p0q ` α˚ OE p0q where pi : E ˆ E Ñ E are the projections and α : E ˆ E Ñ E is the difference map αpx, yq “ x ´ y. Hence we have for ϕΞ : E ˆ E Ñ E ˆ E pϕOE p0q α ϕΞ “ pp1 ϕOE p0q p1 ` pp2 ϕOE p0q p2 ` α ˙ ˙ ˆ ˙ ˆ ˙ ˆ ˆ 2 ´1 1 ´1 0 0 0 0 “ ` ` “ ´1 2 ´1 1 0 1 0 1 Similarly, if Ψ denotes theˆpolarization defined by the divisor E ˆ t0u ` ˙ 2 1 t0u ˆ E ` A, then ϕΨ “ . This implies 1 2 ˆ ˙ˆ ˙ 2 1 2 ´1 ϕΨ ϕΞ “ “ 3 ¨ 1EˆE . 1 2 ´1 2 272

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p Since both polarizations are of type (1,3), Proposition 4.1 gives Ψ “ Ξ. Let C be a smooth projective curve and pJ, Θq its canonically principally polarized Jacobian variety. 4.3 Proposition. For a morphism ϕ : C Ñ X the following statements are equivalent p i) pϕ˚ q˚ Θ ” L

ii) ϕ˚ rCs “ rLs in H 2 pX, Zq. Both conditions imply that ϕ is birational onto its image. Here rCs denotes the fundamental class of C in H 2 pC, Zq. Similarly rLs denotes the first Chern class of L in H 2 pX, Zq. Proof. Identify J “ Jˆ via ϕθ . Condition i) is equivalent to x˚ ϕ˚ ϕLˆ “ ϕpϕ˚ q˚ Θ “ ϕ

By the Universal Property of the Jacobian ϕ extends to a homomorphism from JpCq to X also denoted by ϕ. According to [L-B] Corollary 11.4.2 p Ñ JpCq and ϕ : JpCq Ñ X are related by 235 the homorphisms ϕ˚ : X ˚ p “ ´ϕ . Hence ϕLp “ ϕp ϕ ϕ. Let δpϕpCq, Lq and δpL, Lq denote the endomorphisms of X associated to the pairs pϕpCq, Lq and pL, Lq induced by the intersetion product (see [L-B] Section 5.4). Applying [L-B] Propositions 11.6.1 and 5.4.7 condition i) is equivalent to δpϕpCq, Lq “ ´ϕp ϕϕL “ ´ϕLp ϕL “ ´d ¨ 1X “ ´

pL2 q 1X “ δpL, Lq. 2

According to [L-B] Theorem 11.6.4 this is equivalent to rϕpCqs “ rLs. Since L is of type p1, dq and hence primitive, i) as well as ii) imply that ϕ is birtional onto its image. Hence ϕ˚ rCs “ rϕpCqs.

5 Abelian Surfaces of Type (1,3) Recall that A0p1,3q Ă Ap 1, 3q is the opemn subset corresponding to abelian surfaces X of type p1, 3q with an isogeny onto a Jacobian of 273

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a smooth curve of genus 2. In this section we derive some properties of the elements of A0p1,3q .

Let π : pX, Lq Ñ pY, Pq » pJpHq, OpHqq be an element of A0p1,3q corresponding to the cyhclic ’etale 3-fold covering f : C Ñ H of a smooth curve H of geneus 2 (see Proposition 1.1). According to Proposition 2.1 the Galois group of the composed covering C Ñ H Ñ P1 is the dihedral group D3 generated by an involution j : C Ñ C over the hyperelliptic involution ι of H and a covering transformation τ of F : C Ñ H. According to Proposition 2.2 the involutions j, jτ, jτ2 are elliptic. Denote by fν : C Ñ Eν “ C{ jτν the corresponding coverings. The automorphisms j and τ of C extend to automorphisms of the Jacobian JpCq which we also denote by j and τ. For any point c P C we have an embedding αc : C Ñ JpCq,

p ÞÑ OC pp ´ cq.

Since the double coverings fν : C Ñ Eν are ramified, the pull back homomorphism Eν “ Pic0 pEν q Ñ Pic0 pCq “ JpCq is an embedding (see [L-B] Proposition 11.4.3). We always consider the elliptic curves Eν as abelian subvarieties of JpCq. Then the followind diagram commutes αc

C ❄

❄❄ ❄❄ ❄ fν ❄❄

236

Eν

/ JpCq ③③ ③③ ③ ③ ν ③ } ③ p1` jτ q

(1)

for ν “ 0, 1, 2 and any c P C. Since τp1 ` jτν q “ p1 ` jτν`1 qτ, the automorphismτ of JpCq restritcs to isomorphisms τ : Eν Ñ Eν`1 for ν P Z{3Z. The curve C is containde in teh abelian surface X and generates X as a group, since L “ OX pCq is simple. So the Universal Property of the Jacobian yields a surjective homomorphism JpCq Ñ X, the Kernel of which is described by the following 274

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275

p1 `τp2

5.1 Proposition. 0 Ñ Enu ˆ Eν ÝÝÝÝÑ JpCq Ñ X Ñ 0 is an exact sequence of abelian varieties for ν P z{3Z. Here pi : Eν ˆ Eν Ñ Eν denotes the i-th projection for i “ 1, 2. Forthe proof of the proposition we need the following 5.2 Lemma . For a general c P C either h0 pOC p2c ` jc ` jτcqq “ 1 or h0 pOc p2c ` jc ` jτ2 cqq “ 1. Proof. According to Castelnuovo’s inequality (see [ACGH] Exercies C-1 p.366) C is not hyperelliptic. We indentify C with its image in P3 under the canonical embedding. Assume h0 pOC p2c ` jc jτcqq “ h0 pOC p2c ` jc jτ2 cqq “ 2 for all c P C. Denote Pc “ spanp2c, jc, jτcq and P1c “ spanp2c, jc, jτ2 cq in P3 . According to the Geometric RiemannRoch Theorem (see [ACGH] p.12) the assumption is equivalent to dim Pc “ dim P1c “ 2. For any p P C denote by T pC that tangent of C at p in P3 . Applying the Geometric Riemann-Roch Theorem again, we obtain 4 ´ dimspanpT cC, T jcCq “ h0 pOc p2c ` 2 jcqq ě h0 pOE0 p2π0 pcqqq “ 2. Since h0 pOC p2c`2 jcqq ě 2 by Clifford’s Theorem, dim spanpT cC, T jcCq “ 2. On the other hand, since a general c P C is not a ramification point of a trigonal pencil, we have h0 pOc p2c ` jcqq “ 1 and thus dim spanpT cC, jcq “ 2. Hence Pc “ spanpc, jc, T cCq “ spanpc, jc, T jcCq “ P jc . Similarly we obtain P1c “ P1jc . Since deg C “ 6, this implies Pc X C “ t2c, 2 jc, jτc, τ2 cu

and

P1c X C “ t2c, 2 jc, jτ2 c, τcu.

In particular Pc “ spanpT cC, T jcCq “ P1c . But then the plane Pc contains more than 6 points of C, a contradiction.

275

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Proof of the Proposition. It suffice to prove the proposition for ν “ 0. Step I: The map p1 ` τp2 is injective. According to Lemma 5.2 we may assume h0 p2c ` jc ` jτcq “ 1 (if h0 p2c ` jc ` jτ2 cq “ 1, then we work with ν “ 2 instead of ν “ 0). We have to determine the points p, q P C satisfying the quation p1 ` jqαc ppq ` τp1 ` jqα jτ2 c pqq “ 0. Here we use the fact that E0 “ p1` jqαc pCq “ p1` jqα jτ2 c pCq according to diagram (1). Since h0 p2c ` jc ` jτcq “ 1, the above equation is equivalent to teh following identityh of divisor on C. p ` jp ` τq ` τ jq “ 2c ` jc ` jτc. But the only solution are pp, qq P tpc, τ2 cq, pc, jτ2 cq, p jc, τ2 cq, p jc, jτ2 cqu, all of which represent the point p0, 0q P E0 ˆ E0 “ p1 ` jqαc pCq ˆ p1 ` jqα jτ2 c pCq. Hence p1 ` τp2 is an injective homomorphism of abelian varieties. Step II: The sequence is exact at JpCq. The homomorphism JpCq Ñ X fits into the following commutative diagram JpCq

Nf

❈❈ ❈❈ ❈❈ ❈❈ ! X ④ ④ ④ ④ ④④ g ④ }④

(2)

JpHq

Where N f is the divisor norm map associated to f : C Ñ H and g is an isogeny of degree 3. Since the kernel of N f consists of 3 connected components,the kernel of JpCq Ñ X is an abelian surface. Hence it suffices to show that N f pp1 ` τp2 qpEν ˆ Eν q “ 0. But N f p1 ` jτν q “ p1 ` ττ2 qp1 ` jτν q 276

277

Moduli Spaces of Abelian Surfaces with Isogeny “ p1 ` τ ` τ2 ` j ` jτ ` jτ2 q

is the divisor norm map of the covering C Ñ P1 and hence is the zero map. The Proposition implies that the images of Eν ˆ Eν in JpCq coincide for ν “ 0, 1, 2. Therefore it suffices to consider the case ν “ 0. The automorphism τ of JpCq is of order 3 and induces the identity on JpHq . So by diagram (2) it induces the indentity on X. Hence there is an automorphism T of E0 ˆ E0 of order 3 fitting into the following commutative diagram 0

/ E0 ˆ E0 p1 `τp2/ JpCq

/0

/X

/ 0.

τ

T

0

/X

/ E0 ˆ E0 p1 `τp2/ JpCq

ˆ ˙ 0 ´1 5.3 Lemma. T “ 1 ´1

238

Proof. Since τ is a covering transformation of the 3-fold covering f : C Ñ H, it satisfies the equation τ2 τ ` 1 “ 0 on im pE0 ˆ E0 q Ă ker N f Ă JpCq. So in terms of matrices we have p1 ` τp2ˆ“ p1, τq ˙“ 0 ´1 is p1, ´1 ´ τ2 q. An immediate computation shows that T “ 1 ´1 the only solution of the equation p1, τqT “ τp1, ´1 ´ τ2 q. The automorhism T restricts to isomorphisms T

T

T

E0 ˆ t0u Ý Ñ t0u ˆ E0 Ý Ñ∆Ý Ñ E0 ˆ t0u where ∆ denotes the diagonal in E0 ˆ E0 . Hence p1 ` τp2 maps the curves E0 ˆ t0u, t0u ˆ E0 and ∆ onto E0 , E1 and E2 respectively. 5.4 Lemma. The canonical principal polarization Θ on JpCq induces a polarization Ξ of type p1, 3q on E0 ˆ E0 which is invariant with respect to the action of τ. Moreover Ξ “ rE0 ˆ t0u ` t0u ˆ E0 ` ∆s is H 2 pE0 ˆ E0 , Zq. 277

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Proof. The dual of the divisor norm map N f is the pull back map f ˚ (see [L-B] 11.4(2)). So dualizing diagram (2) above we get JpCq

O a❇❇ ❇❇ Ă ❇❇ ❇❇

f˚

p

=X ⑤⑤ ⑤ ⑤ ⑤⑤ ⑤⑤ pg

JpHq

Denote by P the canonical principal polarization of JpHq. Since p f ˚ qΘ ” 3P and p g is an isogeny of degreee 3, the induced polarization ι˚ Θ on p is of type (1,3). According to Proposition 3.1 and [L-B] ProposiX tion 12.1.3 pE x ˆ E0 , p xq is a pair of complementary abelian subvarieties of JpCq. Hence by [L-B] Corollary 12.1.5 the induced polarization Ξ :“ pp1 ` τp2 q˚ Θ on E0 ˆ E0 is also of type (1,3). moreover Ξ is invariant under T , since the polarization Θ is invariant undert τ. It remains to prove the last assertion. It suffices to prove the equation in the N’eron-Severi group. NSpE0 ˆ E0 q is a free abelian group generated by rE0 ˆt0us, rt0uˆ E0 s, r∆s, and, if E0 admits complex multiplication, also the class rΓs of the graph Γ of an endomorphism γ of E0 . Since Ξ is invariant under T and T permutes the curves E0 ˆ t0u, t0u ˆ E0 and∆, we have Ξ “ aprE0 ˆ t0us ` rt0u ˆ E0 s ` r∆sq ` brΓs 239

for some integers a, b. Assume b ‰ 0. Then necessarily rΓs is invariant with respect to the action of T . Since pt0u ˆ E0 ¨ Γq “ 1, this implies that also 1 “ pE0 ˆ t0u ¨ γq “ p∆ ¨ Γq. on the other hand pE0 ˆ t0u ¨ Γq “ deg γ and p∆ ¨ Γq “ number of fixed points of γ. But on an elliptic curve there is no automporphims with exactly one fix point, a contradiction. So b “ 0. 278

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279

Since the polarization Ξ is type (1,3) and thus 6 “ pΞ2 q “ a2 pE0 ˆ t0u ` t0u ˆ E0 ` ∆q2 “ 6a2 , this implies the assertion.

If we identity JpCq and E0 ˆ E0 with their dual abelian varieties, the map pp1 ` τp2 q^ is a surjective homomorphism JpCq Ñ E0 ˆ E0 . The composed map α

pp1 `τp2 q^

c JpCq ÝÝÝÝÝÝÑ E0 ˆ E0 ac : C ÝÑ

is called the Abel-Prym map of the abelian subvariety E0 ˆE0 of pJpCq, Θq. p “ rE0 ˆ t0u ` t0u ˆ E0 ` As is the dual Recall from Example 4.2 the Ξ polarization of Ξ. 5.5 Lemma. ac : C Ñ E0 ˆ E0 is an embedding and it image ac pCq p defines the polarization Ξ.

Proof. According to [L-B] Corollary 11.4.2 we have a˚c “ α˚c pp1 ` τp2 q “ ´pp1 `τp2 q : EˆE Ñ JpCq. Hence pa˚c q˚ Θ “ pp1 `τp2 q˚ Θ “ Ξ. So Proposition 4.3 implies that αc is birational onto its image and p It remains to show that ac pCq is smooth. But by the ac˚ rCs “ Ξ. adjunction formula pa pac pCqq “

p2 q pΞ 2

` 1 “ 4 “ pg pCq.

p is the kernel of the isogeny ϕ p : E0 ˆE0 Ñ E0 ˆE0 . Recall that KpΞq ˆ ˙ Ξ 2 1 According to Example 4.2 ϕΞp “ and hence 1 2 p “ tpx, xq P E0 ˆ E0 : 3x “ 0u. KpΞq

Consider the dual Tp of the automorphism T of EˆE as an automorphism of E ˆ E. From Lemma 5.3 we deduce that ˆ ˙ 0 1 p T“ ´1 ´1 279

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p is invariant under Tp, i.e. Tp˚ Ξ p “ Ξ. p This Moreover the polarization Ξ follows for example from ´1 ´1 p ˚p p ϕΞp “ 3ϕ´1 p T “ ϕ p Ξ. Ξ “ 3ϕT 2˚ Ξ “ 3T ϕΞ T “ T ϕΞ T

Denote by Fix Tp the set of fixed points of Tp. We obviously have p “ tpx, xq P E0 ˆ E0 : 3x “ 0u. Fix Tp “ KpΞq

p is the set of those points y if E0 ˆE0 , for which This shows that KpΞq the translation map ty commutes with Tp. This is the essential argument in the proof of the following

5.6 Lemma. |E0 ˆ t0u ` t0u ˆ E0 ` A| is the unique linear system p on which induces the identity, i.e. Tp restricts defining the polarization Ξ to an automorphims of every divisor in |E0 ˆ t0u ` t0u ˆ E0 ` A|.

p We first claim that Proof. Let D any divisor defining the polarization Ξ. in Tp˚ D “ D, and the eigenvalue of the corresponding action on the sections defining D is 1, then Tp induces the identity on the linear system p “ FixpTpq we have Tp˚ ty˚ D “ ty˚ D “ ty˚ D. The |D|. For any y P KpΞq Stone-vonNeumann Theorem implies that translating D by elements of p leads to a system of generators of the projective space |D|. So Tp KpΞq acts as the identity on the linear system |D|. This proves the claim. In order to show that |E0 ˆ t0u ` t0u ˆ E0 ` A| is the only linear p on which Tp acts as the identity, note first that Tp˚ pE0 ˆ system defining Ξ t0u`t0uˆE0 `Aq “ E0 ˆt0u`t0uˆE0 `A. Moreover the eigenvalue of the corresponding action the section defining E0 ˆ t0u ` t0u ˆ E0 ` A p contains a divisor of the is 1. Any linear system on E0 ˆ E0 defining Ξ ˚ form tz pE0 ˆ t0u ` t0u ˆ E0 ` Aq for some z P E0 ˆ E0 . Since no group of translations acts on the divisor E0 ˆ t0u ` t0u ˆ E0 ` A itself, the divisor tz˚ pE0 ˆ t0u ` t0u ˆ E0 ` Aq is invariant under Tp if and only if p This completes the proof of the lemma. z P Fix Tp “ KpΞq.

280

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281

5.7 Lemma. a) For any c P C there exists a point y “ ypCq P E0 ˆ E0 uniquely p such that ty˚ ac pCq P |E0 ˆ t0u ` t0u ˆ determined modulo KpΞq E0 ` A|.

˚ a pCq|x P KpΞqu p does not depend on the choice of c. b) The set tty`x c

Proof.

a) According to Lemma 5.5 the divisor ac pCq is algebraically equiva- 241 lent to E0 ˆ t0u ` t0u ˆ E0 ` A. Hence there is a y P E0 ˆ E0 such that the divisors ty˚ ac pCq and E0 ˆ t0u ` t0u ˆ E0 ` A are linp follows from the early equivalent. The uniqueness of y modulo KpΞq p definition of KpΞq.

b) Varying c P C, the subset tty˚ ac pCq|c P Cu of |E0 ˆ t0u ` t0u ˆ E0 ` A| depends continuously on c. On the other hand, the curves in tty˚ ac pCq|c P Cu differ only by translations by elements of the finite p Hence the set tt˚ ac pCq|x P KpΞqu p is independent of group KpΞq. y`x the point c.

6 The Moduli Space A0

p1,3q

In this section we use the results of §5 in order to give an explicit description of the modulo space A0p1,3q .

Let pX, L, πq be an element of A0p1,3q with corresponding e´ tale 3-

fold covering tC Ñ Hu P C32 . The hyperelliptic covering H Ñ P1 lifts to an elliptic covering C Ñ E. The elliptic curve E is uniquely determined by C Ñ E. The elliptic curve E uniquely determined by p denote the polarization on E ˆ E defined by the divisor C Ñ H. Let Ξ E ˆ t0u|t0u ˆ E ` A. According to Lemma 5.5 and 5.7 there is an embedding C ãÑ e ˆ E uniquely determined module translations by p whose image is contained in the linear system |E ˆ element s of KpΞq 281

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t0u `ˆt0u ˆ E ˙ ` A|. On the other hand, consider the automorphism 0 1 Tp “ of E ˆ E. According to Lemma 5.6 it restricts to an ´1 ´1 automorphism of C which coincides with the covering transformation p this implies C X KpΞq p “ H. of C Ñ H. Since Fix Tp “ KpΞq, Let M denote the moduli space of pairs pE, Cq with E an elliptic curve and C a smooth curve in the linear system |E ˆ t0u ` t0u ˆ E ` A| p such that C X KpΞq p “ H. modulo translations by elements of KpΞq Using level structures it is easy to see that M exists as a coarse mouli space for this moduli problem. Summing up we constructed a holomorphic map ψ : A0p1,3q Ñ M. 6.1 Theorem. ψ : A0p1,3q Ñ M is an isomorphism of algrbraic varieties. 242

Proof. It remains to construct an inverse map. Let pE, Cq P M. According to Lemma 5.6 the automorphism Tp of E ˆ E acts on every curve of the linear system. In particular Tp restricts to an automorphism τ of C which is of order 3, since C generates Eˆ as ga group. Moreover τ is fixed point free, since C X Fix Tp “ H. So τ induces an e´ tale 3-fold covering C Ñ H corresponding to an element pX, L, πq P A0p1,3q . It is easy

to see that the map M Ñ A0p1,3q , pE, Cq ÞÑ pX, L, πq is holomorphic and inverse to ψ. 6.2 Corollary. Ap1,3q is a rational variety.

Proof. It suffices to show that the open set M0 “ tpE, Cq P M : E admits no nontrivial automorphismsu is rational. The opent set U “ C ´t0, 1728u parametrizing elliptic curves without nontrivial automorphisms admit a universal family E Ñ U. Consider the line bundle qEˆU E pE ˆU t′u ` t′u ˆU E ` Aq on the fibre product p : E ˆU E Ñ U where A denotes the relative antidiagonal. According to Grauert’s Theorem p˚ OEˆU E pE ˆU t0u ` t0u ˆU E ` Aq is a vector bundle of rank 3 over U. The corresponding projective bundle PU :“ Ppp˚ OEˆU E pE ˆU t0u ` t0u ˆU E ` Aqq parametrizes the linear systems |E ˆ t0u ` t0u ˆ E ` A|. By construction M0 is an open 282

Moduli Spaces of Abelian Surfaces with Isogeny

283

subset of the quotient PU {KpΞq, where KpΞq acts as usual on the fibres of PU Ñ U. Since every vector bundle on U is trivial, PU » P2 ˆ U and PU {KpΞq » P2 {pX{3Z ˆ Z{3Zq ˆ U, which is rational by L¨uroth’s theorem (see [G-H] p. 541).

References [ACGA] Arbarello E., Cornalba M., Griffiths P.A., Harris J., Geometry of algebraic curves I, Spinger Grundlehren 267 (1985). ¨ [C-W] Chevalley C., Weil A.,Uber das Verhalten der Integrale er¨ ster Gattung bei Automorphismen des FunktionenkOrpers, Hamb. Abh. 10(1934) 358–361. [EGAI] Grothendieck A., Dieudonn´e J., El’ements de G´eometrie Alg´ebrique I, Springer Grundlehren 166 (1971). [G] Grothendick A., Let Groupe de Brauer III. In: Dix Expos´es sur la Cohomologie des Sch´emas, North Holland, Amsterdam, (1968) 88–188. [G-H] Griffiths P., Harris J., Principles of Algebric Geometry, John Wiley & Sons, Now York (1978). [H-W] Hulek K., Weintraub S.H., Bielliptic Abelian Surfaces, Math. 243 Ann. 283 (1989) 411–429. [I]

Ingrisch E., Automorphismengruppen und Moduln hyperelliptischer Kurven, Dissertation Erlangen (1985).

[L-B] Lange H., Birkenhakel Ch., Complex Abelian Varieties, Springer Grundlehren 302 (1992). Ch. Birkenhake H. Lange Mathematishes Institut Bismarckstraße 1 21 D-8520 Erlangen 283

Instantons and Parabolic Sheaves M. Maruyama

Introduction 245

S. Donaldson [1] found a beautiful bijection between the set of marked S Uprq-instantons and the set of couples of a rank-r vector bundle on P2C and a trivialization on a fixed line. Then, based on a fixed line. Then, based on Hulek’s result in [3], he concluded that the moduli space of marked S Uprq-instantons with fixed instanton number is connected. Hulek’s result is, however, insufficient to deduce the connectedness. In fact, a vector bundle E on P2C is said to be s-stable in Hulek’s sense if H 0 pP2C , Eq “ 0 and H 0 pP2C , E _ q “ 0. Hulek [3] proved that the set of s-stable vector bundles on P2C wit rpEq “ r, c1 pEq “ 0 and c2 pEq “ n is parametrized by an irreducible algebric set. There are vector bundles on P2C that correspond to marked S Uprq-instatons but are not s-stable. For example, if c2 pEq ă rpEq, E cannot be s-stable and we have, on the other hand, S Uprq-instantons with instanton number n ă r. In this article we shall show that we can regard the couple pE, Hq of a vector bundle E on P2C and a trivialization h of E on a fixed line as a parabolic stable vector bundle. Then, the connectedness of the moduli space of marked S Uprq-instantons reduces to that of the moduli space of parabolic stable sheaves. It is rather complicated but not hard to prove the connectedness of the modulo space of parabolic stable sheaves. The author hopes that he could prove in this way the connectedness of the moduli space of marked S Uprq-instantons. Notation. For a field k and integers m, n, Mpm, n, kq denotes the set of pm ˆ nq-matrices over k and Mpn, kq does the full matrix ring Mpn, n, kq, If f : x Ñ S is a morphism of schemes, E is coherent sheaf on X and if s is a point (or, geromtric point) of S , then Epsq denotes the sheaf E bOS kpsq. For a coherent sheaf F on a variety Y, we denote the rank 284

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of F by rpFq. Assuming Y to be smooth and quasi-projective, we can define the i-th chern class ci pFq of F.

1 A result of Donaldson 246

We shall here reproduce briefly the main part of Donaldson’s Work [1]. Fix a line ℓ in P2C . Let E be vector bundle of rank r on P2C wiht the following properties: (1.1.1) E|ℓ » O‘r ℓ , c1 pEq “ 0 and c2 pEq “ n.

(1.1.2)

(1.1.1) implies that E is µ-semi-stable and hence c2 pEq “ n ě 0. Since the mu-semi-stability of E implies H 0 pP2bC , Ep´1qq “ 0, we see that E is the cohomology sheaf of monad t

s

Ñ L bC OP2 p1q, Ñ K bC OP2 Ý H bC OP2 p´1q Ý where H, K and L are C-vector spaces of dimension n, 2n ` r and n, respectively. s is represented by a p2n ` rq ˆ n matrix A whose entries are linear forms on P2C . Fixing a system of homogeneous coordinates px : y : zq of P2C , we may write A “ A x x ` Ay y ` Az Z, where A x , Ay , Az P Mp2n ` r, n, Cq. Similarly t is represented by B “ D x x ` By y ` Bz z with Bx , By mBz P Mpn, 2n ` r, Cq. The condition ts “ 0 is equivalent to Bx A x “ By Ay “ Bz Az “ 0, Bx Ay “ ´By Az “ ´Bz Ay and Bz A x “ ´Bx Az . E is represented by such a monad uniquely up to the action of GLpHq ˆ GLpKq ˆ GLpLq. We may assume that ℓ is defined by z “ 0. Then E|ℓ is trivial if and only if det Bx Ay ‰ 0. We can find bases of H, K and L so that Bx Ay “ In , 285

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where In is the identitiy matrix of degree n. Changing the basis of L, we have ¨ ˛ ¨ ˛ 0 n In n A x “ ˝ 0 ‚n, Ay “ ˝In ‚n 0 r 0 r

and

n n r

n

Bx “ p0, In , 0q, 247

Then, setting

n r

By “ p´In , 0, 0q.

¨ ˛ α1 ˝ A z “ α2 ‚ a

with α1 , α2 P Mpn, Cq and a P Mpr, n, Cq, the equations By Az “ ´Bz Ay and Bz A x “ ´Bx Az imply Bz “ p´α2 , α1 , bq with b P Mpn, r, Cq. The last equation Bz Az “ 0 means 1.2 rα1 , α2 s ` ba “ 0. On ℓ we have an exact sequence u

v

0 Ñ Oℓ p´1q Ý Ñ O‘2 Ñ Oℓ p1q Ñ 0. ℓ Ý The restriction of our monad to ℓ is ‘n u‘0

‘n v‘0

‘r Oℓ p´1q‘n ÝÝÝÝÑ p‘n O‘2 ÝÝÝÑ Oℓ p1q‘n ℓ q ‘ Oℓ Ý

and the trivialization of E|ℓ comes from the last term of the middle. The equivalence defined by the action of GLpHq ˆ GLpKq ˆ GLpLq induces an action of the group $ ¨ ´1 ˇ , ˛ ˇ p 0 0 & . ˇ p´1 0‚, pqˇˇ p P GLpn, Cq, q P GLpr, Cq pp, ˝ 0 % ˇ 0 0 q on the above normalized matrices. The action of q is nothing but changing the trivializations of E|ℓ . 286

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The condition that the above normalized couplepA, Bq gives rise to a vector bundle is the for every pλ, µ, νq P P2C , ˛ λIn ` να1 λA x ` µAy ` νAz “ ˝µIn ` να2 ‚ νa ¨

is an injection of H to K and λBx ` µBy ` µBz “ p´µIn ´ να2 , λIn ` να1 , νbq is a surjection of K to L. If ν “ 0, then these conditions are trivially satisfied. Thus we have the following. 1.3 Proposition. The set tpE, hq|E has the properties (1.1.1) and (1.1.2). h is a trivilization of E|ℓu { » is in bijective correspondence with the set of quadruples pα1 , α2 , a, bq of matrices with the following properties (1.3.1), (1.3.2) and (1.3.3) modulo an action of GLpn, Cq: α1 , α2 P Mpn, Cq, a P Mpr, n, Cq and b P Mpn, r, Cq,

(1.3.1) 248

rα1 , α2 s ` ba “ 0,

(1.3.2)

f or all pλ, µq P C2 , ˛ ¨ λIn ` α1 ˝µIn ` α2 ‚ a

(1.3.3)

in injective and p´µIn ´α2 , λIn `α1 , bq is surjective. Here p P GLpn, Cq sends pα1 , α2 , a, bq to ppα1 p´1 , pα2 p´1 , ap´1 , pbq. Let us embed P2C into P3C as the plane defined by w “ 0, where px : y : z : wq is a system of homogeneous coordinates of P3C . Now we look at the vector bundle defines by a monad c

d

Ñ K bC OP3 Ý Ñ L bC OP3 p1q, H bC OP3 p´1q Ý 287

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and trivialized on ℓ “ tz “ w “ 0u, c and d are represented by A “ A x x ` Ay y ` Az z ` Aw w B “ Bx x ` By y ` Bz z ` Bw w

with

and

¨ ˛ In ˝ A x “ 0 ‚, 0

Bx “ p0, In , 0q,

¨ ˛ 0 ‚ ˝ I Ay “ n , 0

¨ ˛ α1 ˝ Az “ α2 ‚, a

By “ p´In , 0, 0q,

¨ ˛ αˆ 1 ˝ α Aw “ ˆ 2 ‚ aˆ

Bz “ p´α2 , α1 , bq,

ˆ Bw “ pαˆ2 , αˆ1 , bq.

The condition dc “ 0 of the monad means rα1 , α2 s ` ba “ 0,

(1.4.1)

ˆ a “ 0, rαˆ1 , αˆ2 s ` bˆ

(1.4.2)

ˆ “ 0. rα1 , αˆ2 s ` rαˆ1 α2 s ` bˆa ` ba

(1.4.3)

249

Let H “ R ` Ri ` R j ` Rk be the algebra of quaternions over R. Regarading C4 as H2 , bH acts on C4 from left. Hence we have a j-invariant real analytic map π : P3C “ tC4 zt0uu{C˚ Ñ S 4 – P2H . A vector bundle E of rank r on P3C is called an instanton bundle if it comes from as S Uprq-instanton on S 4 , or equivalently if the following conditions are satisfied: there is and isomorphism λ of E to j˚ pE _ q such that the composition j˚ pt λ´1 q. λ is equal to idE when we identify j˚ p j˚ pE _ q_ q with E, (1.5.1) E|π´1 pxq is trivial for allx P S 4 . 288

(1.5.2)

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Instantons and Parabolic Sheaves

Ab instanton bundle E is the cohomology of a monad which appeared in the above. The instanton structure of E provides this space K with a _ Hermitian structure and an isomorphism of L to H . For λ “ pλ1 , λ2 , λ3 , λ4 q P C4 , we set Apλq “ A x λ1 ` Ay λ2 ` Az λ3 ` Aw λ4

bpλq “ Bx λ1 ` By λ2 ` Bz λ3 ` Bw λ4 _

Since Apλq˚ “t Apλq defines a linear map of K – K to H have a linear map

_

– L, we

Apλq “ Apλq˚ ‘ Bpλq : K Ñ L ‘ L. The quaternion algebra H acts on L ‘ L by ˆ ˙ ˆ ? ˙ ˆ ˙ ˆ ˙ ´ ´1u u u v ? i , j “ “ v v ´v ´ ´1v Then the condition that Apqλqpvq “ qApλqpvq

for all q P H, v P K

is equivalent to the condition that pA, Bq gives rise to an S Uprq-instanton bundle. Now the above condition can be written down in the form ˙ ˆ t ´ Az w `t Aw z Ap jp0, 0, z, wqq “ ´Bz w ` Bw z ˙ ˆt ˙ ˆ Bz z ` Bw w Az z `t Aw w “ j “ Bz z ` Bw w ´t Az z ´t Aw w that is,

250 t

Aw “ Bz and t Az “ ´Bw .

Thus we have αˆ1 “ ´α˚2 , αˆ 2 “ α˚1 , aˆ “ b˚ and bˆ “ ´a˚ . On the other hand, Apλq is injective if and only if Apλq˚ is surjective. Thus our 289

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monad defines a vector bundle if and only if Apλq˚ is sujective. Thus our monad defines a vector bundle if and only if Apλq is surjective for all λ P C4 zt0u. (1.3.3) implies that Apλq is surjectifve for λ P C3 zt0u which gives our plane w “ 0 and then so is Apqλq. Since tqλ|q P H, λ P C3 zt0uu sweeps C4 zt0u, (1.3.3) is equivalent to the condition that Apλq is surjective for all λ P C4 zt0u Now the condition (1.4.2) is the adjoint of (1.4.1) or (1.3.2) and the condition (1.4.3) becomes rα1 , α˚1 s ` rα2 , α˚2 s ` bb˚ ´ a˚ a “ 0. Let MpS Uprq, nq be the set of isomorphism classes of marked S Uprqinstantons with instanton number n. MpS Uprq, nq is the set of the isomorphism classes of teh couples p∇, gq of an S Uprq-instanton ∇ and an element g of the fiber over northpole of the S Uprq-principle fiber bundle where the instantion is defined. What we have seen in the above is stated as follows. 1.6 Proposition. MpS Uprq, nq is in bijective correspondence with the Upnq-quotient of the set of quadruples tpα1 , α2 , a, bqu of matrices with the properties (1.3.1), (1.3.2), (1.3.3) and rα1 , α˚1 s ` rα2 , α˚2 s ` bb˚ ´ a˚ a “ 0.

(1.6.1)

Assume that G “ GLpn, Cq acts on CN with a fixed norm structure such that Upnq does isometrically. Take a G-invariant subscheme W of CN whose points are att stable. Let W0 be the set points that are nearest to the origin in its G-orbit. 1.7 Proposition. W{G is isomorphic to W0 {Upnq. Let us look at the C-linear space V “ Mpn, CqˆMpn, CqˆMpr, nCqˆ Mpn, r, Cq where GLpn, Cq acts as in Proposition 1.3. ThenUpnq acts on V isometrically with respect to the obvious norm of V. Let W be the subscheme of V defined by (1.3.2) and (1.3.3). Then we have have the following key results. 1.8 Lemma. [1, p. 458] A point pα1 , α2 , a, bq in W is contained in W0 if and only if it has the property (1.6.1). 290

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1.9 Lemma. [1, Lemma]. W is contained in the set of stable points of V with respect to the action of GLpn, Cq. Therefore, we come to the main result of [1]. 1.10 Theorem. The set tpα1 , α2 , a, bq | p1.3.1q, p1.3.2q and p1.3.3q 251 are satisfiedu {GLpn, Cq is in bijective correspondence with the set tpα1 , α2 , a, bq|p1.3.1q, p1.3.2q and p1.1.2q. h is a trivialization of E|ℓ u{ – is isomorphic to the space MpS Uprq, nq .

2 Parabolic sheaves Let px, OX p1q, Dq be a triple of a non-singular projective variety X over an algebraically closed field k, an ample line bundle OX p1q on X and an effective Cartier divisor D on X. A coherent torsion free sheaf E is said to be a parabolic sheaf if the following data are assigned to it: a filtration 0 “ Ft`1 Ă Ft Ă ¨ Ă F1 “ EbOX OD by coherent subsheaves, (2.1.1) a system of weights 0 ď α1 ă α2 ă ¨ ¨ ¨ ă αt ă 1. (2.1.2)

We denote the parabolic sheaf by pE, F˚ , α˚ q. For a parabolic sheaf pE, F˚ , α˚ q, we define par ´ χpEpmqq “ χpEp´Dqpmqq `

t ÿ

i“1

αi χpFi {Fi`1 pmqq.

If E 1 is a coherent shubsheaf of E with E{E 1 torsion free,then we have an induced parabolic structure. In fact, since E 1 bOX OD can be regarded 1 Ă Fδ1 Ă as a subsheaf of E bOX OD , we have a filtration 0 “ Fδ`1 Fδ1 Ă ¨ ¨ ¨ Ă Fδ1 “ E 1 bOX OD such that F 1j “ E 1 bOX OD X Fi for some i. The weight α1j of F 1j is defined to be αi with i “ maxtk|F 1j “ E 1 bOX OD X Fk u.

2.2 Definition. A parabolic pE, F˚ , α˚ q is said to be stable if for every coherent subsheaf E 1 with E{E 1 torsion free and with 1 ď rpE 1 q ă rpEq and for all sufficiently large integers m, we have par ´ χpE 1 pmqq{rpE 1 q ă par ´ χpEpmqq{rpEq, 291

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where the parabolic structure of E 1 is the induces from that of E. Let S be a scheme of finite type over a universally Japanese ring and now let pX, OX p1q, Dq be a triple of a smooth, projective, geometrically integral scheme X over S , an S -ample line bundle OX p1q on X an effective relative Cartier divisor D on X over S . 252

2.3 Lemma. If T is a locally noetherian S -scheme and of E is aT -flat coherent sheaf on X ˆ s T such that for every geometric point t of T , Eptq is torsion free, then E|D “ E bOX OD is flat over T . Proof. For every point y of T , Epyq is torsion free and Dy is a Cartier divisor on Xy . Thus the natural homomorphism Ep´Dqpyq Ñ Epyq is injective. Then, since E is T -flat, E|D “ E|Ep´Dq is T flat Q.E.D. Let E be a coherent sheaf on X ˆS T which satisfies the condition in the above lemma. E is said to be a T -family of parabolic sheaves if the following data are assigned to it: a filtration 0 “ Ft`1 Ă Ft Ă ¨ ¨ ¨ Ă F1 “ E|D by coherent

(2.4.1)

a system of weights 0 ď α1 ă α2 ă ¨ ă αt ă 1.

(2.4.2)

subsheaves such that for 1 ă i ă t, E|D {Fi is flat over T,

As in teh absolute case we denote by pE, F˚ , α˚ q the family of parabolic sheaves. For T -families of parabolic sheaves pE, F X,α˚ q and pE 1 , F˚1 , α1˚ q, they are said to he equivalent and we denote pE, F˚ , α˚ q „ pe1 , F˚1 , α1˚ q if there is an invertible sheaf L on T such that E is isomorphic to E 1 bOT L, the filtration F˚ is equal to F˚1 bOT L under this isomorphism and if the systems of weights are the same. Fixing polynomials Hpxq, H1 pxq, . . . Ht pxq and a system of weights 0 ď α1 ă ¨ ¨ ¨ ă αt ă 1, we set ˇ $ , ˇpE, F˚ , α˚ qis aT ´ family of / ’ ’ ˇ / & ˇ parabolic sheaves with the . L par ´ ΣpH, H˚ , α˚ qpT q “ pE, F˚ , α˚ q ˇˇ „ ’ ’ / ˇ the properties p2.5.1q and / % ˇ p2.5.2q

for every geometrix point y of T and 1 ď i ď t, χpepyqpmqq “ (2.5.1) 292

293

Instantons and Parabolic Sheaves Hpmq and χppEpyq{Fi`1 pyqqpmqq “ Hi pmq,

for every geometric point y of T, pEpyq, F˚ pyq, α˚ q is stable. (2.5.2) Obviously par ´ ΣpH, H˚ , α˚ q defines a contravariant functor of the category pSch {S q of locally notherian S -schemes to that of sets. Note that for every geometric point s of S and every member pE, F˚ , α˚ q P par ´ ΣpH, H˚ , α˚ qpsq, we have par ´ χpEpmqq “ Hpmq ´

t ÿ

i“1

εi Hi pmq,

where εi “ αi`1 ´ αi with αt`1 “ 1. 253 One of main results on parabolic stable sheaves is stated as follows. 2.6 Theorem. [7] Assume that all the weights α1 , . . . , αt are rational numbers. Then the functor par ´ ΣpH, H˚ , α˚ q has a coarse moduli scheme MX{S pH, H˚ , α˚ q of locally of finite type over S . If S is a scheme over of field of characteristic zero, then the coarse moduli scheme is quasi-projective over S . Now let us go back to the situation of the preceding section. We have a fixed line ℓ in P2C , a torsion free, coherent sheaf E of rank r on P2C and a trivialization of E|ℓ . There is a bijective correspondence between the set of trivilzations of E|ℓ and the set „

TE “ tϕ : E|ℓ Ñ Oℓ pr ´ 1q|H 0 pϕq : H 0 pE|ℓ q Ý Ñ H 0 pOℓ pr ´ 1qqu{ – where – means isomorphism as quotient sheaves.

2.7 Lemma. Let E be a torsion free, coherent sheaf of rank r on P2C such that E|ℓ is a trivial vector bundle. For every element pϕ : E|ℓ Ñ Oℓ pr ´ 1qq of TE , kerpϕq is isomorohic to Oℓ p´1q‘r´1 Proof. Since kerpϕq is a vector bundle of rank r ´ 1 on the line ℓ, it is isomorphic to a direct sum Oℓ pa1 q ‘ ¨ ¨ ¨ ‘ Oℓ par´1 q of line bundles. Our condition on ϕ means that H 0 pℓ, kerpϕqq “ 0, Combining this and 293

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the fact that degpkerpϕqq “ 1 ´ r, we see that a1 “ ¨ ¨ ¨ “ ar´1 “ ´1. Q.E.D. Fixing a system of weights α1 “ 1{3, α2 “ 1{2, every element ϕ of TE gives rise to a parabolic structure of E: 0 “ F3 Ă F2 “ kerpϕq – Oℓ p´1q‘r´1 Ă F1 “ E|ℓ . 2.8 Proposition. If E as in Lemma 2.7. Assume E has the properties (1.1.1) and (1.1.2), then the above parabolic sheaf is stable. Proof. Set r “ rpEq. We know that pm ´ 1q2 3pm ´ 1q n ` `1´ 2 2 r pr ´ 1qm pr ` mq ` ` 2r ˆ ˙ 3r 1 1 n m2 ` 1´ m` ´ . “ 2 6r 3 r

par ´ χpEpmqq{rpEq “

254

Pick a coherent subsheaf E 1 of E with E{E 1 torsion free and write par ´ χpE 1 pmqq{rpE 1 q “ Then we see a1 “ µpE 1 q `

m2 ` a1 m ` a0 . 2

1 1 ´s t¯ ` ` , 2 rpE 1 q 2 3

where s ` t “ rpE 1 q and 0 ď t ď 1. Thus if µpE 1 q ă 0, then we have a1 ď

1 ´1 1 `1ă1´ `qă r 6r rpE 1 q

and hence we obtain the desired inequality. We may assume therefore that c1 pE 1 q “ 0. Since E{E 1 is torsion free, E 1 is locally free in a neighborhood of ℓ and E 1 |ℓ is a subsheaf of E|ℓ . Thus the triviality of E|ℓ ‘rpE 1 q and hence E 1 |ℓ is not contained in F2 . Since implies tha E 1 |ℓ – Oℓ 294

Instantons and Parabolic Sheaves

295

E 1 |ℓ {pF2 X E 1 |ℓ q is subsheaf of F1 {F2 – Oℓ pr ´ 1q, it is of rank 1. We have two cases. Case 1. F2 X E 1 |ℓ “ 0 and rpE 1 q “ 1. Then the length of the filtration of E 1 |ℓ is 1 and the weight is 1/3. Thus we see that a1 “

1 1 1 1 ` “1´ ă1´ . 2 3 6 6r

Case 2. F2 X E 1 |ell ‰ 0. In this case F21 “ F2 X E|1ℓ is a subsheaf of Oℓ p´1q‘pr´1q and hence for m ě 0, χpF21 pmqq ď prpE 1 q ´ 1qm. On the other hand, since E 1 |ℓ {F21 is a subsheaf of Oℓ pr ´ 1q, we have that for m ě 0, χpE 1 |ℓ {F21 pmqq ď m ` r. Combining these, we get a1 ď

1 1 1 1 1 1 ` ´ ` “1´ ă1´ . 1 1 1 2 2 2rpE q 3rpE q 6r 6rpE q

This completes our proof.

3 Connectedness of the moduli of instantons 255

Let us set rx2 3rx ` `r´n 2 2 H1 pxq “ x ` r Hpxq “

H2 pxq “ rx ` r 1 1 α1 “ and α2 “ . 3 2

295

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M. Maruyama

For these invariants, we have the modulo space Mpr, nq “ MpH, H˚ , α˚ q of parabolic stable sheaves on pP2C , OP2 p1q, ℓq. There is an open sub˜ nq consisting of pE, F˚ , α˚ q with the properties scheme Mpr, nq of Mpr, E|ℓ – O‘r ℓ and E|ℓ pr ´ 1q, 0 for the surjection ϕ : E|ℓ Ñ E|ℓ {F2 , Hpϕq is isomorphic.

(3.1.1) (3.1.2)

Mpr, nq contains a slightly smaller open subscheme Mpr, nq0 consisting of locally free sheaves. What we have seen in the above is MpS Uprq, nq – Mpr, nq0 . 3.2 Proposition. Mpr, nq is smooth and of pure dimension 2rn. To prove this proposition we shall follow the way we used in [5]. Let us start with the general setting in Theorem 2.6. Let Σ be the family of the classes of parabolic stable sheaves on the fibres of X over X with fixed invariants. For simplicity we assume that Σ is bounded (Proposition 3.2 is the case). If m is a sufficiently large integer and if pE, F˚ , α˚ q is a representative of a member of Σ over a geometric point s of S , then we have that both Ep´D s qpmq and Epmq are generated by their global sections (3.3.1) and for all i ą 0, H i pEp´D s qpmqq “ H i pEpmqq “ 0, for 1 ď j ď t and i ą 0, H i pF j {F j`1 pmqq “ 0 and F j pmq is

(3.3.2)

generated by its global sections. Replacing every member

pE, F˚ , α˚ q P par ´ ΣpH, H˚ , α˚ qpT q by pOX pmq bOS E, OX pmq bOS F˚ , α˚ q, we may assume m “ 0. Setting N “ dimH 0 pXpsq, Eq for a member pE, F˚ , α˚ q of Σ on a fiber X s over 256 s, fixing a free OS -module V of rank N and putting VX “ V bOS OX , there is an open subscheme R of Q “ QuotVHX {X{S such that for every 296

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Instantons and Parabolic Sheaves

algebraically closed field k,a k-valued point z of Q is in Rpkq if and only if the following conditions are satisfied: For the universal quotient ϕ : VX bOS OQ Ñ E˜ the induced ˜ the induced map H 0 pϕpZqq : V bOs kpsq Ñ H 0 pX s , EpZqq

(3.3.3)

is an isomorphism, wheresis the image ofzin S pkq.

˜ For every i ą 0, H i pXpsq, Epzqq “ 0. ˜ Epzq is torsion free

(3.3.4) . (3.3.5)

˜ Then, We denote the restriction of E˜ to XR “ X ˆS R by the same E. ˜ DR is flat over R. Let Rt be the R-scheme by Lemma 2.3 we see that E| QuotE| and let ˜ D {XR {R R

˜ DR bOR ORt Ñ E˜ t E| be the universal quotent. Assume that we have a sequence R j Ñ R j`1 Ñ ¨ ¨ ¨ Ñ Rt Ñ R ˜ DR such that of scheme R j and a sequence of sheaves E˜ j , E˜ j`1 . . . , E˜ t , E| E˜ i is Ri -flat coherent sheaf on XRi and that there is a surjection E˜ i bORi ORi´1 Ñ E˜ i´1 . Set R j´1 to be QuotE˜ j {XR

j

{R j

and take the universal quotient

E˜ j bOR j OR j´1 Ñ E˜ j´1 . on R j´1 . By induction on j we come to R1 and we have a sequence of surjections ˜ DR bOR OR1 Ñ E˜ t bOR OR1 Ñ ¨ ¨ ¨ Ñ E˜ 2 bOR OR1 Ñ E˜1 . E| t 2 Setting E “ E˜ bOR OR1 297

298

M. Maruyama ˜ DR b OR OR1 Ñ E˜ i bOR OR1 q Fi “ kerpE| i

we get an R1 -family of parabolic sheaves. There is an open subscheme U of R1 such that for every algebraically closed field k, a k-valued point z of R1 is in Upkq if and only if pEpxq, F˚ pzq, α˚ q is stable. We shall denote the restriction of pE, F˚ , α˚ q to U by the same pE, F˚ , α˚ q. 257 The S -group scheme G “ GLpVq naturally acts on Q and R is Ginvariant. There is a canonical G-linearization on z˜ and hence G so acts on Rt that the natural morphism of Rt to R is G-invaraiant. Then E˜ t carries a natural G-linearization. Tracing these procedure to R1 , we come eventually to a G-action on U and a G-linearization of the family pE, F˚ , α˚ q. Obviously the center Gm,s of G acts trivially on U and we have an action of G “ G{Gm,s . We can show that there exists a geomeric quotient of U by G. Then we see The quotient U{G is the moduli scheme in Theorem 2.6.

(3.4)

For a T -family of parabolic sheaves pE, F˚ , α˚ q, we put Ki “ kerpE ´ E|Dr {Fi q and then we get a sequence of T -flat coherent subsheaves Kt`1 “ Ep´DT q Ă kt Ă ¨ ¨ ¨ Ă K2 Ă K1 “ E. For a real number α, there is an integer i such that 1 ď i ď t ` 1 and αi´1 ă α ´ rαs ď αi , where α0 “ αt ´ 1 and αt`1 “ 1. Then we set Eα “ Ki p´rαsDT q. Thus we obtain a filtration tEα uαPR of E parametrized by real numbers that has the following properties E{Eα isT ´ flat and if α ď β, then Eα Ě Eβ .

(3.5.1)

if ε is a sufficiently small positive real number, then Eα´ε “ Eα .. (3.5.1) For every real number α, we have Eα`1 “ Eα p´DT q. 298

(3.5.3)

Instantons and Parabolic Sheaves

299

E0 “ E.

(3.5.4)

The length of the filtration for 0 ď α ď 1 i f f inite.

(3.5.5)

Convesely, if E is a T -flat coherent sheaf on XT such that for every geometric point y of T , Epyq is torsion free that E has a filtration parametrized by R with the above properties, then we have a T -family of parabolic sheaves. Thus we may use the notation E˚ for a T - family of parabolic sheaves pE, F˚ , α˚ q. 3.6 Definition. Let E and E˚1 be T -familes of parabolic sheaves. A homomorphism f : E Ñ E of the undelying coherent sheaves is said to be a homomorphism of paraboic sheaves if for all 0 ď α ă 1, we have f pEα q Ă Eα1 . HomPar pE˚ , E˚1 q denotes the set of all homomorphisms of parabolic 258 sheaves of E˚ to E˚1 . If one notes that for a stable parabolic sheaf E˚ on a projective varaiety, a homomorphism of E˚ to itself is the multiplication by an element of the ground field k or HomPar pE˚ , E˚ q – k, then one can prove the following lemma by the same argument as the proof of Lemma 6.1 in [5]. 3.7 Lemma. Let A be an artinian local ring with residuce field k and let E˚ be a SpecpAq-family of parabolic sheaves. Assume that the restriction E “ E˚ bA k to th closed fiber is stable. Then the natural homomorphism A Ñ HomPar pE˚ , E˚ q is an isomorphism. Let us go back to the situation of (3.4). Replacing the role of Lemma 6.1 argument of Lemma 6.3 of [5] by the above lemma, we get a basic result on the action of G in U. 3.8 Lemma. The action of G on U is free. It is well-known that this lemma implies the following (see [5], Proposition 6.4). 3.9 Proposition. The natural morphism π : U Ñ W “ U{G is a principal fiber bundle with group G 299

300

M. Maruyama Now we come to our proof of Proposition 3.2.

Proof of Proposition 3.2. There is universal space U whose quotient ˜ nq “ MpH, H˚ , α˚ q under the notatior before by the group G is Mpr, ˜ nq Proposition 3.2. Since our moduli space Mpr, nq is open in Mpr, we have a “ G-invariant open subsheme P of U which mapped onto Mpr, nq. Tlanks to Proposition 3.9 the natural quotient morphism π : P Ñ Mpr, nq is a principal fiber bundle with group G and hence we have only to show that P is smooth and has the right dimension. Put X “ P2C . Fix an integer m which satisfies the conditions (3.3.1) and (3.3.2) for our Mpr, nq. We set Hrmspxq “ Hpx ` mq. Take a point E of Mpr, nqpCq and a C-vector space V of dimension N “ H 0 pX, Epmqq. we have an surjection θ : V x “ V bC OX Ñ Epmq. Since the kernel K of θ is locally free, HomOx pK, Epmqq is the tangent Hrms space of Q “ QuotV {X{C at the point q that is given by the above sex

quence and an obstraction of the smoothness of Q at q is in H 1 pX, K _ bOX Epmqq. Since Ext2OX pE, Eq is dual space of HomOX pE, Ep´3qq, it van259 ishes for stable E. We can apply the same argument as ion the proofs of Propositions 6.7 and 6.9 in [5] to our situation and we see that Q is smooth and of dimension 2rn ´ r2 ` N 2 at the point q. P is a folber space over an open subscheme of Q whose fibers are an open subscheme of QuotO‘r {ℓ{C consisting of surjections ℓ

O‘r ℓ Ñ Oℓ pr ´ 1q such that the induced map of global sections in bijective. By Lemma 2.7 the space of obstructions for the smoothness of P over Q is H 1 pℓ, Oℓ prq‘r´1 q “ 0. Thus P is smooth over Q and hence so is over SpecpCq. Moreover, the dimension of the fibers is equal to 0 dim H pℓ, Oℓ prq‘r´1 q “ r2 ´ 1. Combining this and the above result 300

Instantons and Parabolic Sheaves

301

on the dimension of Q, we see that dimP “ 2rm ` N 2 ´ 1. Since dim G “ N 2 ´ 1, Mpr, nq is of dimension 2rn at every point. Q.E.D. Base on Proposition 3.2 and Hulek’s result stated in Introduction, we can prove the following. 3.10 Theorem. Mpr, nq0 is connected. Proof. Our proof is divided into several steps. We set as befor X to be P2C . (I) If r “ 1, then Mpr, nq is the moudli space of ideals with colenght n which define 0-dimentsional closed subshcemes in Xzℓ. It is well known that this is irreducible [2]. (II) Assume that r ě 2. By Hulek’s result we see that Up0q “ tE˚ P Mpr, nq0 |H 0 px, Eq “ H 0 pX, E _ q “ 0u is irreducible. In fact, we have a surjective morphism of a PGLprqbundle over Hulek’s parameter space of s-stable bundles to Up0q. Let us set Upaq “ tE˚ P Mpr, nq0 | dim H 0 pX, Eq “ au, 1 Upa, bq “ tE˚ P Upaq|E – O‘b X ‘ E 1 , E 1 ﬂ OX ‘ E 1 u.

Then Upaq is locally closed and Upa, bq is constructible in Mpr, nq0 . (III) We shall compute the dimension of Upa, bq. For an E˚ P Upa, bq, 260 there is an extension 0 Ñ O‘a X Ñ E Ñ J Ñ0 because E is µ-semi-stable and c1 pEq “ 0[[6], Lemma 1.1]. We have moreover that J|ℓ – O‘a x is a direct summand of E and put E “ O‘b ‘ E . Consider the exact sequence 1 X 0 Ñ O‘a´b Ñ E1 Ñ J Ñ 0. X 301

302

M. Maruyama For the double dual J 1 of J, we set T “ J 1 {J and c “ dim H 0 px, T q. Then we see 2 pT, OX q – ExtO1 X pJ, OX q ExtO x and dim H 0 px, ExtO2 X pT, OX qq “ c. On the other hand, since c2 pJ _ q “ n ´ c, J _ is locally free and since H 2 pX, J _ q “ 0, we have dim H 1 px, J _ q “ n ´ c ´ pr ´ aq ` d, where d “ dim H 0 pX, J _ q. Thus we see ‘pa´bq

dim Ext1OX pJ, OX

q “ dim H 1 pX, pJ _ q‘a´b q`

dim H 0 pX, ExtOt x pJ, OX q‘a´b q

“ pa ´ bqtn ´ pr ´ aq ` du.

If we change free bases of O‘a´b , then we obtain the same sheaf. X 1 Hence if dim ExtOX pJ, OX q ă a ´ b, then every extension of J by O‘a´b contians OX as a direct factor, which is not the case. X Therefore, we get a ´ b ď n ´ pr ´ aq ` d

or

n´r`b`d ě0

The extensions of J by O‘a´b are now parametrized by a space of X dimension pa ´ bqtn ´ pr ´ aq ` du and each point of the space is contained in a subspace of dimension pa´bq2 `dim EndOX pJq´1 whose points parmetrize the same extension. (IV) Let us fix a system of homogeneous coordinates px0 : x1 q of ℓ and r´q identity J|ℓ with the free sheaf ‘i“1 Oℓei . If we have a surjection ϕ : J|ℓ “ ‘r´a i“1 Oℓei Ñ Oℓ pr ´ a ´ 1q 261

and if f1 px0 , x1 q “ ϕpe1 q, . . . , fr´a px0 , x1 q “ ϕper´a q are liearly independent, then for general homogeneous forms g1 , . . . , gr´a of 302

303

Instantons and Parabolic Sheaves

degree a ´ 1 and gr´q`1 , . . . , gr of degree r ´ 1, we define a map ϕ˜ of ‘ of ‘ri“1 Oℓei to Oℓ pr ´ 1q by ϕpe ˜ i q “ gi x0r´a ` x1a fi

ϕpe ˜ jq “ g j

1ďiďr´a

r ´ a ` 1 ď j ď r.

Choosing g j suitably, we obtain a surjective ϕ˜ : ‘ri“1 Oℓei Ñ Oℓ pr ´ 1q which induces a bijection between the spaces of global sections. Conversely, if we have a homomorphism ψ of ‘ri“1 Oℓei to Oℓ pr ´ 1q such that h1 “ ψpe1 q, . . . , hr “ ψper q are linearly independent. The we can write hi uniquely in the following way: hi “ h1i x0r´a ` x1a h”i where h1i (or, h2j ) is of degree a ´ 1 (or , r ´ a ´ 1, resp.). There is a permutations σ of t1, . . . ru such that h2σp1q , . . . , h2σpr´aq are linearly independent. Thus, after a permutation of indices, the above procedure produces ψ from a homomorphism J|ℓ Ñ Oℓ pr ´a´1q which induces a bijection between the spaces of global sections. We see therefore that the parabolic structures on E are parametrtized by a fiber space over the space of parabolic structures of J whose fibres are a finite union of open subschemes of an affine space of dimension 2ra ´ a2 . (V) Every element of EndQX pEq induces an endomorphism of the space H 0 pX, Eq and hence gives rise to an element of EndQX pJq. Let End J pEq denote the subspace of EndQX pEq consisting of the elements which induce the identity on J. Then we have dim End J pEq ě dim EndOX pEq ´ dim EndOX pJq. On the other hand, we see ‘b EndOX pEq “ EndOX pO‘b X q ‘ HomOX pOX , E 1 q

‘ HomOX pE1 , O‘b X q ‘ EndOX pE 1 q. 303

304

M. Maruyama Since dim H 0 pX, J _ q “ d, there is an exact sequence 0 Ñ J1 Ñ J Ñ G Ñ 0 with pG_ q_ – O‘d X . For a ξ in ‘dpa´bq

q – H 0 px, OX HomOX pG, O‘a´b X

q,

we have a member of EndpE1 q ξ

E1 Ñ J Ñ G Ý Ñ O‘a´b Ñ E1 . X 262

Thus dim EndOX pE1 q ě dpa ´ bq ` 1. Therefore, we get the following inequality dim End J pEq ě dim EndOX pEq ´ dim EndOX pJq

ě b2 ` bpa ´ bq ` ab ` dpa ´ bq ` 1 ´ dim EndOX pJq “ ab ` ad ` 1 ´ dim EndOX pJq

(VI) There is a couple pA, J˜˚ q of a scheme A and an A-family J˜˚ of parabolic sheaves which parametrizes all parabolic stable sheaves J˚ with rank r ´ a and c2 pJq “ n such that the restriction J|ℓ is trivial vector bundle and H 0 pX, Jq “ 0. We may assume that A is reduces and quasi-finite over Mpr ´ a, nq, and hence dim A ď 2npr ´ aq. By Proposition 2.8 the sheaf J defined in the step (III) appears as the underlying sheaf of a parabolic sheaf parametrized by J˜˚ . For the underlying sheaf J˜ of the family J˜˚ , we have a resolution by a locally free sheaves 0 Ñ B1 Ñ B0 Ñ J˜ Ñ 0. Using this resolution and splitting out A into the direct sum of suitable subschemes, we can construct a locally free sheaf C on A such that for every point y of A, we have a natural isomorphism ˜ Cpyq – Ext1OX p Jpyq, OX q. 304

Instantons and Parabolic Sheaves

305

Note that C is not necessarily of constant rank. On D “ Vppc_ q‘a´b q we have a universal section Ξ of the sheaf g˚ pC ‘a´b q, where g : X ˆ D Ñ X ˆ A is the natural projection. Let ξi be the projection of ξ to the i-the direct factor of C ‘a´b . The subset D0 ty P D|ξ1 pyq, . . . , ξa´b pyq span a linear subspace of rank a ´ b Cpyqu ˚ ˜ ˜ 0 Ñ O‘a´b XˆD0 Ñ E 1 Ñ g p Jq Ñ 0, where we denote g|D0 by g. Let H be the maximal open subscheme of D0 where E˜1 is locally free. Set E˜ “ E˜ 1 |XˆH ‘ O‘b XˆH . (III) tells us dim D0 ď 2npr ´ aq ` pa ´ bqtn ´ pr ´ aq ` du. Note here that d may depend on connected components of D0 . Furthermore, by the result of (III) again, each point of H is contained. in a subspace of dimension pa ´ bq2 ` dim EndOX pJq ´ 1 where E˜ 1 parmetrizes the same extensions. (VII) By breaking up H into the direct sum of suitable subschemes, we 263 ˜ ℓˆH has a constant trivialization and hence may assume that g˚ p Jq| ˜ ℓˆH . By the result of (IV), the parabolic structure of g˚ p Jq ˜ so is E| ˜ provides us with a fiber space p : Z Ñ H and a Z-family E˚ of parabolic stable vector bundles such that pX, E˜ ˚ q parametrizes all the parabolic stable sheaves contained in Upa, bq and that every fiber of p is of dimension 2ra ´ a2 . Thus we see that dim Z ď 2npr ´ aq ` pa ´ bqtn ´ pr ´ aq ` du ` 2ra ´ a2 “ 2nr ´ na ´ nb ` ra ` rb ´ ab ` ad ´ bd

Moreover, the conclusion of (V) shows that on the fiber of p, each point is contained in a closed subsheme of dimension at least ab` ad ` 1 ´ dim EndOX pJq whose points define the same parabolic sheaf. (VIII) The family E˜ ˚ gives rise to a morphism of Z to Mpr, nq0 whose image is exactly our Upa, bq. Combining the results of (VI) and 305

306

M. Maruyama (VII) we get dim Upa, bq ď 2nr ´ na ´ nb ` ra ` rb ´ ab ` ad ´ bd ´ pa ´ bq2 ´ dim EndOX pJq ` 1 ´ ab ´ ad ´ 1 ` dim EndOX pJq “ 2nr ´ pn ´ r ` aqa ´ pn ´ r ` b ` dqb

Since for every member E˚ of Upa, bq, the underlying sheaf E is µ-semi-stable, Riemann-Roch implies n ´ r ` a “ dim H 1 pX, Eq ě 0. This and the inequality we obtained in (III) show that dim Upa, bq ď 2nr. Replacing E by E _ in the definition of Upaq and Upa, bq, we define U _ paq and U _ pa, bq. Then, by the same argument as above we come to a family pZ 1 mE˚1 q of the dual bundles of the members of U _ pa, bq. By taking the dual basis of the trivial sheaf E 1 |ℓˆZ 1 , we have an isomorphism E 1_ |ℓˆZ 1 Ñ E 1 |ℓˆZ 1 . Combining this isomorphism and the parabolic structure of E˚1 , we obtain 1 a Z 1 -family E˚_ of parabolic sheaves which parametrizes all the members of U _ pa, bq. The dimension of U _ pa, bq is the same as Upa, bq. (IX) Assume that n ě r. In this case we see that ˜ ¸ ˜ ď ď ď ď Mpr, nq0 “ Up0q Upa, bq aě1,bě0

aě1,bě0

¸

U _ pa, bq .

By our result in (VIII) we see that if a ď 1, the both dim Upa, bq and dim U _ pa, bq are less than 2rn. On the other hand, we know that Up0q is irreducible by [3]. This completes the proof of this case.

306

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(X) Assume that n ă r . Then, Riemann-Roch implies that for every 264 member E˚ of Mpr, nq0 , we have dim0 pX, Eq ě r ´ n. Thus we see that ˜ ¸ ď ď MpR, nq0 “ Upr ´ nq Upa, bq . aąr´n,bě0

As in (IX) if a ą r ´ n, then dim Upa, bq ă 2rn. This means that it is sufficient to prive that Upr ´ nq is connected. For a member E˚ of Upr ´ nq, there is an exact sequence 0 Ñ O‘r´n Ñ E Ñ J Ñ 0. X According to the type of J, Upr ´ nq is deivided into three subschemes: V0 “ tE˚ P Upr ´ nq|Jis locally free, H 0 pX, J _ q “ 0u

V1 “ tE˚ P Upr ´ nq|Jis locally free, H 0 pX, J _ q ‰ 0u V2 “ tE˚ P Upr ´ nq|Jis locally freeu.

For J of E˚ P V0 , we have that H 0 pX, J _ q “ 0, c1 pJ _ q “ 0, c2 pJ _ q “ n, rpJ _ q “ n and J _ is µ-semi-stable. Hence RiemannRoch implies that Ext1OX pJ, OX q “ H 1 pX, J _ q “ 0. Then V0 is contained in Upr ´ n, r ´ nq or the undelying sheaf E of a member of V0 is written in a form O‘r´n ‘ J with J s-stable. Since these X J’s are parametrized by an irreducible variety, so are the members of V0 . This proves the irreducibilty of V0 . Applying the argument before (VIII) to the set tJ _ |E˚ P V1 u, we find that tJ˚ |E˚ P V1 u is of dimension less the 2n2 . Then the dimension of A in (VI) for V1 for V1 is of dimension less the 2n2 and hence dim V1 ă 2nr. A similar argument tells us that our remaining problem is to prove that L “ tJ˚ |E˚ P V2 u is of dimension less than 2n2 . 1

(XI) For the underlying sheaf J of a member J˚ of L, we set J to be the double dulal of J. Since J is locally free in a neighborhood 307

308

M. Maruyama of ℓ, the parabolic structure of J˚ induces of J 1 . J 1 {J is a torsion sheaf supported by a 0-dimensional subscheme of X. L is the disjoint union of L1 , . . . , Ln , where Lm “ tx1 , . . . xk u be the support of J 1 {J. If the length of the artinian module pJ 1 {Jq xi is ai , piq piq then there is a filtration 0 Ă T ai Ă ¨ ¨ ¨ Ă T 1 “ pJ 1 {Jq xi such piq

piq

p1q

that T j {T j`1 – kpxi q. Let J j J1

265

Ñ

p1q p1q T 1 {T j`1

p2q Jj

and

be the kernel of the sujection p1q

pkq

pkq

we can define a filtration Jak Ă ¨ Ă J1 p1q

piq

p2q

p2q

be the kernel of Ja1 Ñ T 1 {T j`1 . Thus piq

p1q

Ă ¨ ¨ ¨ Ă Ja1 Ă ¨ ¨ ¨ Ă piq

pi`1q

Ă J 1 such that J j {J j`1 – kpxi q and Jai {J1

– kpxi`1 q pi´1q and we see that “ J. Since Jai´1 pXi q is an n-dimensional pi´1q vector space, the surjection of Jai´1 to kpxi q is parametrized by an pn ´ 1q-dimensional projective space. Since TorO 1 pkpxi q, kpxi qq is isomorphic to kpxi q‘2 , the exact sequence J1

pkq Jak

piq

piq

TorO 1 pkpxi qq, kpxi q Ñ J j pxi q Ñ J j´1 pxi q Ñ kpxi q Ñ 0 piq

piq

shows us that dim J j pxi q ď dim J j´1 pxi q ` 1. Therefore, surpiq

jections of J j to kpxi q is parametrized by a projective space of dimension less than or equal to n ` j ´ 2, where O “ OX,xi . Fixing J˚1 tJ˚ P Lm |pJ _ q_ – J 1 u is parametrized by a space of dimemsion less than or equal to δm “

ai k ÿ ÿ

pn ` j ´ 2q ` 2k

i“1 j“1

k ÿ ai pai ´ 3q ` 2k 2 i“1 ¸ ˜ k ÿ a2i 3ai ´ `2 . “ nm ` 2 2 i“1

“ nm `

On the other hand, the space tJ˚1 |J˚ P Lm u is of dimension 2npn´ mq. Therefore Lm is of dimension at most 2npn ´ mq ` δm . Now 308

309

Instantons and Parabolic Sheaves we have ˜

¸ a2i 3ai 2npn ´ mq ` δm “ 2n ´ nm ` ´ `2 2 2 i“1 ¸ ˜ k ÿ a2i 3ai 2 2 ´ `2 ď 2n ´ m ` 2 2 i“1 ¸ ˜ k 2 ÿ ÿ a 3a i i ` ´ 2 ` ai aj . “ 2n2 ´ 2 2 j‰1 i“1 k ÿ

2

It is easy to see that k ÿ

i“1

˜

ÿ a2i 3ai ` ´ 2 ` ai a j 2 2 j‰i

¸

is non-negative and equal to 0 if and only if n “ 1. If n “ 1, then both V0 and V1 are empty and V2 is exactly tJ “ I x with I x the ideal of a point x P Xzℓu. There is a unique locally free sheaf G x which is an extension of I x by O x . Finally we see that in this 266 case the set undelying sheaves of the members of Upr ´ nq is tO‘r´2 ‘ G x |x P Xzℓu X which is parametrized by the irreducible variety Xzℓ.

Q.E.D.

Now we come to the connectedness of the moduli space of marked S Uprq-instantons. 3.11 Corollary. The moduli space MpS Uprq, nq of marked S Uprqinstantons with instanton number n is connected.

References [1] S.K. Donaldson, Instantosn and geometric invariant theory, Comm. Math. Phys. 93 (1984) 453-460. 309

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[2] J. Fogarty, Algebraic families on an algebraic surface, Amet. J. Math 90 511-521. [3] K. Hulek, On the classification of stable rank-r vector bundles on thte projective plane, Proc. Nice conference on vector bundles and differential equation, Birkh¨auser (1983) 113-142. [4] G. Kempf and L. Ness, Lengths of vectors in representation spaces, Lect. Notes in Math. 732, Springer-verlag, Berlin-Heidelberg-New York-Tokyo, (1978) 233-243. [5] M. Maruyama, Moduli of stable sheaves II, J. Math. Kyoto Univ. 18 (1978) 557-614. [6] M. Maruyama, On a compatification of a moduli space of stable vector bundles on a rational surface, Alg, Geom. and Comm. Alg. in Honor of Masayoshi NAGATA, Kinokuniya, Tokyo (1987) 233260. 267

[7] M. Maruyama and K. Yokogawa, Moduli of parabolic stable sheaves, to appear in Math. Ann. [8] C. H. Taubes, Pathconnected Yang-Mills moduli spaces, J. Diff. Geom. 19 (1984) 337-392.

Department of Mathmatics Faculty of Sciense Koyoto University Saky-ku, Kyoto 606-01 Japan

310

Numerically Effective line bundles which are not ample V. B. Mehta and S. Subramanian

1 Introduction In [6], there is a construction of a line bundle on a complex projective 269 nonsingular variety which is ample on very propersubvariety but which is nonnample on the ambient variety. The example is obtained as the projective bundle associated to a “general” stable vector bundle of degree zero on a compact Riemann surface of genus g ě 2. Now,w e can show by an an algebraic argument valid in any characteristic, the existence of a variety of dimension ď 3 with a line bundle as above. The details of the proof will appear elsewhere.

2 Let C be a complete nonsingular curve defined over an uncountable algerbaically closed field (of any characteristic). Let Mrs denote the moduli space of stable bundles of rank r and gegree zero on C and Mrss the moduli of semistable bundles of rank r and degree zero. We assume throught that the curve C is ordinary. We can show 2.1 Proposition. Let the characteristic of the ground field be positive and F the frobenius morphism on C. There is a proper closed subset of Mr˚ such that for any stable bundle V in the complement of this closed set, F ˚ V is also stable. Proof. We use Artin’s theorem on the algrbraisation of formal moduli space for proving the above proposition. We have as a corollary to the proof of Proposition (2.1). 311

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2.1.1 Corollary. Let C be an oridinary curve. Then the rational map f : Mrss Ñ Mrss induces by the Frobenius F : C Ñ C, is etale on an open set, and in particular, dominant. 270

2.1.2 Corollary. Let C be an ordinary curve, For any positive integer k, there is a nonempty open subset of Mrs such that for V in this open set, F m˚ V is stable for 1 ď m ď k. Proof. We apply Proposition (2.1) and Corollary (2.1.1) successively. Q.E.D. We have 2.2 Proposition. Given a finite etale morphism p : C1 Ñ C, there eixsts a proper closed subset of Mrs such that any vector bundle in the complement of this closed set remains stable on C1 . We have 2.3 Proposition. For a fixed positive ieteger k, there is a nonempty open subset of Mrs such for any stable bundle V in this open set, there is no nonzero homomorphism from a line bundle of degree zero to S k V. Using the above results, we can show 2.4 Theorem. Let C be nonsigular ordinary curve of genus ě 2 over an uncountable algebrically closed field (of any characteristic). There is a dense subset of Mrs such that for any stable bundle V in this dense set, we have ˚

1) F k pVqis stable for all K ě 1. 2) For any separable finite morphism π : C˜ Ñ C, π˚ pVq is stable. 3) There is no nonzero homomorphism from a line bundle of degree zero to the symmetric power S k pVq for any K ě 1. Remark. If C is a smooth curve defined over a finite field (of characalg teristic p) then any continuous irreducible represenation ρ : π1 pCq Ñ S Lpr, F p q of the algebraic fundamental group of C of rank r over the finite field defines a stable vector bundle V on C such that F m˚ V » V for some m ď 1 (see [4]). Such a bundle V statisfies F k˚ pVq is stable 312

Numerically Effective line bundles which are not ample

313

for all K ě 1. We can construct such representations for any curve C of genus g ě 2 when r is coprime to p, and for an ordinary curve C when p divides r.

3 271

Let C be a nonsingular ordinary curve of genus ě 2, and V a stable vector bundle of rank 3 degree zero on C satisfying the conditions of Theorem (2.4) above. Let π : PpVq Ñ C be the projective bundle associated to V and L “ OPpVq p1q the universal line bundle on PpVq. Then we have 3.1 Theorem. The line bundle L is ample on very proper subvariety of PpVq, but L is not ample on PpVq. Proof. We can check that L.C ą 0 for any integral curve C, and that L2 . D ą 0 for any irreducible divisor D. This implies that L|D is ample on D. This shows that L is ample on divisors in PpVq and hence on any proper subvariety of PpVq. Also, L, is not ample on PpVq. Q.E.D. 3.2 Remark. The case r “ 2 is covered by the first part of Theorem (3.1).

References [1] M. Artin,The implicit function theorem in Algebraic Geometry, Proceedings International Colloquium on Algebraic Geometry, Bombay 1968, Oxford University Press (1969) 13–34. [2] T. Fujita, Semipositive line bundles, Jour. of Fac. Sc. Univ. of Tokyo, Vol. 30 (1983) 353–378. [3] R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes No. 156 (Springer-Verlag) (1970) 313

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[4] H. Lange and U. Stuhler, Vecktor bundel and Kurven und Darstellungender algebraisechen Fundamental gruppe, Math. z. 156 (1977) 73–83. [5] V. B. Mehta and A. Ramanathan, Restriction of stable sheaves and representations of the fundamental group, Inv. Math 77 (1984) 163–172. [6] S. Subramanian, Mumford’s example and a general construction, Proc. Ind. Acad. of Sci. Vol. 99, December (1989) 97–208.

School of Mathematcs Tata Institute of Fundamental Research Homo Bhabha Road, Colaba Bombay 400 005 INDIA.

314

Moduli of logarithmic connections Nitin Nitsure The talk was based on the paper [N] which will, appear elsewhere. 273 What follows is a summary of the results. Let X be a non-singular projective variety, with S Ă X a divisor with normal crossings. A logrithmic connection E “ pE, ∇q on X with sigularity over S is a torsion free coherent sheaf E together with a Clinear map ∇ : E Ñ Ω1X rlog S s b E satisfying the Leibniz rule and having curvature zero, where Ω1X rlog S s is the sheaf of 1-forms on X with logarithmic singularities over S . By a theorem of Deligne [D], a connection with curvature zero on a non-nonsigular quasi-projective variety Y is regular if and only if given any Hironaka completion X of Y (so that X is non-sigular projective and S “ X ´ Y is a divisor with normal crossings), the connection extends to a logrithmic connections on X with singularity over S . Carlos Simpson has constructed in [S] a moduli scheme for nonsingular connections (with zero curvature) on a projective variety. A simple example (see [N]) shows that a modulo scheme for regular connections on a quasiprojective variety does note in general exist. Therefore, we have to consider the moduli problem for loarithmic connections on a projective variety. The main difference between non-singular connections and logarithmic connections is that for logarithmic connections, we have to define a notion of (semi)-stability, and restrict ourselves to these. We say that a logarithmic connection is (semi-)stable, if usual inequality between normalized Hilbert polynomials is satisfied for any ∇-invariant coherent subsheaf. In the case of non-sigular connections on a projective variety, the normalized Hilbert polynomial is always the same, so semistability is automatically fullfilled. Followigng Simpson,s method, with the extra feature of keeping track of (semi-)stability, we prove the exitance of coarse moduli scheme for (S-equivalen classes of) semistable315

316

Nitin Nitsure

logarithmic connections which have a given Hilbert polynomial. We also show that the infinitesimal deformations of a locally free logarithmic connection E are parametrized by the first hypercohomology of the logarithmic de Rham complex associated with End pEq. 274 A given regular connection on a quasi-projective variety Y has infinitely many extensions as logarithmic connections on a given Hironaka completion X of Y. A canonical choice of such an extension is given by the fundamental construction of Deligne [D], which gives a locally free logarithmic extension. Using our description, we show that certain extensions of any given regular connections are rigid, that is, they have no infinitesimal deformations which keep the underlying regular connections fixed. The cirterion for this is that no two distinct eigenvalues of the residue the logarithmic connection must differ by an integer. In paticular, this shows that Deligne’s construction gives a rigid extension.

References [D] Deligne, P., Equations differentielles a points singuliers reguliers, Lect. Notes in Math. 163 Springer (1970) [N] Nitsure, N., Moduli of semi-stable logarithmic connections, Journal of Amer. Math. Soc. 6 (1993) 597–609. [S] Simpson, C., Moduli of representations of the fundamental group of a smooth projective variety, Preprint (1990) Princeton University.

School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400 005 India

316

The Borel-Weil theorem and the Feynman path integral Kiyosato Okamoto

Introduction Let p and q be the canonical mometum and coordinate of a particle. In 275 the operator method of quantization, corresponding ot p and q there are operators, P, Q which in the coordinate representation have the form: ? d A “ q, P “ ´ ´1 . dp The quantization means the correspondence between the Hamiltonian fucntion hpp, qq and the Hamiltonian operator H “ hpP, Qq, where a certain procedure for ordering noncommuting operator arguments P and Q is assmed. The path integral quantization is the?method to compute the kernel function of the unitary operator expp´ ´1tHq. Any mathematically strict definition of the path integral has not yet given. In one tries to compute path integral in general one may encounter the difficulty of divergence of the path integral. Many examples, however, show that the path integral is a very poweful tool to compute the kernel function of the operator explicitly. The purpose of this lecture is to explain what kinds of divergence we have when we try to compute the path integral for complex polarizations of a connected semisimple Lie group which contains a compact Cartan subgroup and to show that we can regularize the path integral by the process of “normal ordering” (cf. Chapter 13 in [10]). The details and proofs of these results are given in the forthcoming paper [7]. Since a few in the audience do not seem to know about the Feynman path integral I would like to start with explaining it form a point of view of the theory of unitary representations, using the Heisenberg group which is most deeply related with the quantum mechanics. 317

318 276

Kiyosato Okamoto

The Feynman’s idea of the path integral can be easily and clearly understood if one computes the path integral on the coadjoint orbits of the Heisenberg group: $¨ , ˛ & 1 p r . G “ ˝ 1 q‚; p, q, r P R . % 1

The Lie algebra of G is given by , $¨ ˛ . & 0 a c g “ ˝ 0 b‚; a, b, c P R . % 0

The dual space of g is identified with , $¨ ˛ . & 0 g˚ “ ˝ ξ 0 ‚; ξ, η, σ P R . % σ η 0

by the pairing

g ˆ g˚ Q pX, λq ÞÝÑ trpλXq P R.

Any nontrivial coadjoint orbit is given by an element ˛ ¨ 0 λσ “ ˝ 0 0 ‚ for some σ ‰ 0 σ 0 0

Then the isotropy subgroup at λσ is given by $¨ , ˛ & 1 0 r . Gλσ “ ˝ 1 0‚; r P R , % 1 and the Lie algebra of Gλσ is $¨ , ˛ & 1 0 c . ˝ ‚ 0 0 ;c P R . gλσ “ % 0 318

The Borel-Weil theorem and the Feynman path integral

319

We consider the real polarization: $¨ , ˛ & 0 a c . p “ ˝ 0 0‚; a, c P R . % 0

Then the analytic subgroup of G corresponding to p is given by $¨ , ˛ & 1 p r . P “ ˝ 1 0‚; p, r P R . % 1

Clearly the Lie algebra homomorphism ¨ ˛ 0 a c ? ? p Q ˝ 0 0‚ ÞÝÑ ´ ´1σc P ´1R. 0

lifts to the unitary character ξλσ : ˛ ¨ 1 p r ? P Q ˝ 1 0‚ ÞÝÑ e´ ´1σr P Up1q. 1

Let Lξλσ denote the line bundle associated with ξλσ over the homogenous space G{P. Then the space C 8 pLξλσ q of all complex valued C 8 -sections of Lξλσ can be identified with ( f P C 8 pGq; Fpgpq “ ξλσ ppq´1 f pgq pg P G, p P Pq .

For any g P G we define an operator πpλσ pgq on C 8 pLξλσ q : For f P C8 pLξλσ q pπpλσ pgq f qpxq “ f pg´1 xq px P Gq.

Let Hλpσ be the Hilbert space of all square integrable sections of Lξλ . Then πpλσ is a unitary representation of G on Hλpσ . We put $¨ , ˛ & 1 0 0 . ˝ ‚ 1 q ;q P R . M“ % 1 319

277

320

Kiyosato Okamoto

Then as is easily seen the product mapping M ˆ P ÝÑ G is a real analytic isomorphism which is surjective. Let f P C 8 pLξλσ q. Then, since ¨ ˛ ¨ ˛ 1 p r 1 p r ? f pg ˝ 1 0‚q “ e ´1σr f pgq for q P G, ˝ 1 0‚ P P, 1 1 f can be uniquely determined by its values on M. From this we obtain the following onto-isometry: Hλpσ Q f ÞÝÑ F P L2 pRq 278

where

˛ 1 0 0 Fpqq “ f p˝ 1 q‚q pq P Rq. 1 ˛ ¨ 0 a c For any g “ exp ˝ 0 b‚ P G, we define a unitary operater Uλpσ pgq 0 on L2 pRq such that the diagram below is commutative: ¨

Hλpσ πpλσ pgq

/ L2 pRq Uλpσ pgq

/ L2 pRq.

Hλpσ Then we have ¨

˛ 1 0 0 pUλpσ Fqpqq “ f pg´1 ˝ 1 q‚q 1 ˛¨ ˛ ¨ 1 0 0 1 ´a ´c ` ab 2 “ f p˝ 1 ´b ‚˝ 1 q‚q 1 1 320

The Borel-Weil theorem and the Feynman path integral

321

¨

˛¨ ˛ 1 0 0 1 ´a ´c ` ab 2 ´ aq ‚q “ f p˝ 1 q ´ b‚˝ 1 0 1 1 “e

?

´1σp´c` ab 2 ´aqq

Fpq ´ bq.

Now we show that the above unitary operator is obtained by the path integral. In the following, for the definition of the connection form θλσ , the şT hamiltonian HY and the action 0 γ˚ α ´ HY dt in the general case the audience may refer to the introduction of the paper [5]. We use the local coordinates q, p, r of g P G as follows: ¨ ˛¨ ˛ 1 0 0 1 p r G Q g “ ˝ 1 q‚˝ 1 0‚. 1 1

Since the canonical 1-form θ is given by g´1 dg, we have

θλσ “ă λσ , θ ą“ trpλσ g´1 dgq “ σpdr ´ pdqq. We choose

279

αp “ ´σpdq. Then

dα f p ´σd p ^ dq “ . 2π 2π For Y P g, the hamiltonian HY is given by HY “ trpλσ g´1 Ygq “ σpaq ´ bp ` cq

˛ ¨ 0 a c where Y “ ˝ 0 b‚. The action is given by 0 żT 0

γ˚ α ´ HY dt “

żT 0

t´σpptqqptq ´ σpaq ´ bp ` cqudt. 321

322

Kiyosato Okamoto

We divide the time interval r0, T s into N-equal small intervals żT 0

˚

γ α ´ HY dt “

N ż ÿ

k“1

k nT k´1 N T

γ˚ α ´ HY dt.

“ k´1 N

T, Nk T

The physicists’ calculation rule asserts that one should take the “ordering”: * N " ÿ qk ` qk´1 T ´σpk´1 pqk ´ qk´1 q ´ σpa ´ bpk´1 ` cq . 2 N k“1 This choice of the ordering can be mathematically formulated as follows. “ ‰ k We take the paths: for t P k´1 N T, N T qptq “ qk´1 ` pt ´

pptq “ pk´1 ,

k ´ 1 qk ´ qk´1 , Tq N T {N

qp0q “ q, and qpT q “ q1 . Then the action for the above path becomes N ż ÿ

k“1

k NT k´1 N T

“

t´σpptqqptq ´ σpaq ´ bp ` cqudt

N " ÿ

k“1

qk ` qk´1 T ´σpk´1 pqk ´ qk´1 q ´ σpa ´ bpk´1 ` cq 2 N

*

The Feynman path integral asserts that the transition amplitude be280 tween the point q “ q0 and the point q1 “ qN is given by the kernel function which is computed as follows: KYp pq1 , q; T q “ lim

ż8

NÑ8 ´8

ˆ exp

#

¨¨¨

ż8 ż8

´8 ´8 N „ ÿ

? ´1σ

k“1

¨¨¨

ż8

´8

|σ|

d pN´1 d p0 ¨ ¨ ¨ |σ| dq1 ¨ ¨ ¨ dqN´1 2π 2π

qk ` qk´1 T ´pk´1 pqk ´ qk´1 q ´ pa ´ bpk´1 ` cq 2 N

322

+

‰

The Borel-Weil theorem and the Feynman path integral ż8

ż8

323

N ź

T δp´qk ` qk´1 ` b q NÑ8 ´8 N ´8 k“1 + # N ÿ ? apqk ` qk´1 q T ` cq ˆ ´ ´1σ p 2 N k“1 " * ? bT “ lim δp´qN ` q0 bT q exp ´ ´1σpaT pq0 ` q ` cT q NÑ8 2 " * 2 ? abT “ δp´q1 ` qbT q exp ´ ´1σpaqT ` ` cT q . 2 “ lim

¨¨¨

dq1 ¨ ¨ ¨ dqN´1

For F P Cc8 pRq we have ş8 p 1 ´8 KY pq , q; T qFpqqdq

˙* ˆ ? abT 2 1 ` cT Fpq1 ´ bT q “ exp ´ ´1σ aq T ´ 2 “ pUλpσ pexp T YqFqpq1 q. "

Thus the path integral gives our unitary operator. In the above quantization, the hamiltonian function HY pp, qq corresponds to ?

´1

? d p d Uλσ pexp tYq|t“0 “ σpaq ` cq ´ ´1b , dt dq

which is slightly different form ? ? d d HY p´ ´1 , qq “ σpaq ` c ` ´1b q. dq dq This difference comes from the fact that we chose αp “ ´σpdq whereas physicists usually take pdq.

1 Coherent representation 281

In this section, we compute the path integral for unitary representations realized by the Borel-Wiel theorem for the Heisenberg group. In other 323

324

Kiyosato Okamoto

words, we compute the path integral for a complex polarization which is called by physicists the path integral for a complex polarization which is called by physicists the path integral for the coheretnt representation. We shall show that the path integral, also in this case, gives unitary operators of these representations. The complexification GC of G and gC of g are given by $¨ , ˛ & 1 p r . GC “ ˝ 1 q‚; p, q, r P C , % 1 $¨ , ˛ & 0 a c . C g “ ˝ 0 b‚; a, b, c P C . % 0 we consider the complex polarization defined by ? $¨ , ˛ ´1b c & 0 . ˝ ‚ 0 b ; a, b, c P C . p“ % 0

We denote by P the complex analytic subgroup of GC corresponding to p. We put W “ GP “ GC . Then it is easy to see that Lie algebra homomorphism ? ˛ ¨ 0 ´1b c ? 0 b‚ ÞÝÑ ´ ´1σc P C pQ˝ 0

lifts uniquely to the holomorphic character ξλλσ : ¨ 1 PQ˝

?

? ˛ ´1b c ` 2´1 b2 ? ‚ ÞÝÑ e´ ´1σc P C˚ . 1 b 1

We denote by Lξλσ the holomorphic line bundle on GC {P associated with the character ξλλσ . 324

The Borel-Weil theorem and the Feynman path integral

325

We denote by ΓpLξλσ q the space of all holomorphic sections of Lξλσ and by ΓpCq the space of all holomorphic functions on C. 282 We use the coordinates of g P G: ? ¨ 0 ´ 2´1 z ˚ ˚ g “ exp ˚ 0 ˝

¨

?

´1 ˚1 ´ 2 z ˚ “˚ ˚ 1 ˝

0

¨

˛

?

´1 2 z

˚0 ´ ‹ ˚ ‹ 1 ‹ exp ˚ ˚ 0 2 z‚ ˝ 0 ˛¨ ? ? ´1 2 z ‹ ˚1 ´ 2´1 z ´ 8 ‹˚ ‹˚ 1 ‹˚ 1 z 2 ‚˝ 1

˛ ? ´1 2 r` |z| ‹ 4 ‹ ‹ 1 ‹ z 2 ‚ 0 ˛ ? ? ´1 2 ´1 2 |z| ` r` z‹ 4 8 ‹ ‹ 1 ‹ z 2 ‚ 1

where z P C, r P R. We have the isomorphism ¯ ´ Γ Lξλλ Q f ÞÝÑP ΓpCq σ

where

¨

?

´1 2 z

˚1 ´ ˚ FpZq “ f p˚ 1 ˚ ˝

˛ ´1 2 ´ z ‹ 8 ‹ ‹q 1 ‹ 2z ‚ 1 ?

square integrable We denotes by Γ2 pLξλσ q the Hilbert´space of all ¯ σ 2 |σ| holomprphic sections of Lξλσ and by Γ2 C, 2π e´ 2 |z|

For any g P G we define an operator πpλσ pgq on Γ2 pLξλσ q : For f P Γ2 pLξλσ q pπpλσ pgq f qpxq “ f pg´1 xq px P Gq. Then πpλσ is a unitary representation of G on Γ2 pLξλσ q. Since ż

G{Gλσ

2

| f pgq| ωλσ

ˇ2 ż ˇ ˇ ´ σ|z|2 ˇ dzdz ˇ 4 FpZqˇ |σ| “ ˇe ˇ 2π C

325

326

Kiyosato Okamoto “

ż

C

|Fpzq|2 e´

σ|z|2 2

|σ|

dzdz , 2π

where we denote by ωλσ the canonical symplectic form on the coadjoint orbit Oλσ “ G{Gλσ and we put ? ´1 dz ^ dz. dzdz “ 2 |σ|

The above isomorphism gives an isometry of Γ2 pLξλσ q onto Γ2 pC, 2π σ 2 e´ 2 |z| q. σ 2 |σ| 283 As is easily seen Γ2 pC, 2π e´ 2 |z| q ‰ t0u if and only if σ ą 0. If follows that Γ2 pLξλσ q ‰ t0u if and only if σ ą 0. For the rest of the section we assume that˘σ ą 0. ` 0 P G, we define a uniFor any g “ exp 0 a c 0 b p σ ´ σ2 |z|2 2 q such that the diagram below tary operator Uλσ pgq on Γ pC, 2π e is commutative: Γ2 pLξλσ q πpλσ pgq

/ Γ2 pC,

Γ2 pLξλσ q Then we have pUλpσ pgqFqpzq ¨ ?

/ Γ2 pC,

σ ´ σ2 |z|2 q 2π e

Uλpσ pgq

σ ´ σ2 |z|2 q 2π e

˛ ´1 2 ´ z ‹ ˚1 ´ 8 ˚ ‹ ‹q 1 “ f pg´1 ˚ 1 z ˚ ‹ 2 ˝ ‚ 1 ¨ ˛ ? ? ´1 2 ´1 ˛ ¨ ˚1 ´ 2 z ´ 8 z ‹ 1 ´a ´c ` ab 2 ˚ ‹ ‹q 1 1 ´b ‚˚ “ f p˝ 1 z ˚ ‹ 2 ˝ ‚ 1 1 ´1 2 z

?

326

The Borel-Weil theorem and the Feynman path integral ¨ ˛ ? ? ´1 ´1 2 ˚1 ´ 2 pz ´ γq ´ 8 pz ´ γq ‹ ˚ ‹ ‹ 1 “ f p˚ 1 pz ´ γq ˚ ‹ 2 ˝ ‚ 1 ? ? ? ¨ ´1 1 ´ 2 γ ´c ` 4´1 |γ|2 ´ 2´1 γz ` ˚ ˚ ˆ˚ 1 ´ γ2 ˝

327

?

´1 2 8 γ

1

? σp´ ´1c´ 14 |γ|2 ` 21 γzq

“e

˛

‹ ‹ ‹q ‚

Fpz ´ γq, ? Where γ “ b ` ´1a. It is well-known that Uλpσ is an irreducible unitary representation σ

2

σ ´ 2 |z| e of G on Γ2 pC, 2π q. Using the parametrization: ˛¨ ¨ ? ? ´1 2 ´1 ˚1 ´ 2 z ´ 8 z ‹ ˚1 ‹˚ ˚ ‹˚ 1 g“˚ 1 z ‹˝ ˚ 2 ‚ ˝ 1

?

´1 2 z

1

r`

?

´1 2 4 |z|

`

1 2z

1

we have

? zdz ´ zdz q, θλσ “ trpλσ g´1 dgq “ σpdr ` ´1 ˆ ˙ 4 ? γz ´ γz HY “ σ ´1 `c . 2 We choose αp “ ´

?

´1σ 2 zdz.

Then we have ? 1 ´1σdz ^ dz dαp “ . 2π 4π ‰ “ k For fixed z, z1 P C we define the paths: For t P k´1 N T, N T zptq “ zk´1 ,

327

?

´1 2 8 z

˛

‹ ‹ ‹, ‚ 284

328

Kiyosato Okamoto ˆ ˙ k´1 zk ´ zk´1 zptq “ zk´1 ` t ´ T , N T {N zp0q “ z and zpT q “ z1 .

Then the action becomes ? ˆ? ˙* żT" ? ´1γzptq ´ ´1γzptq 1 σzptq9zptq ´ ´1σ `c dt 2 2 0 ? ˆ? ˙* N ż kT " ÿ ? N ´1γzptq ´ ´1γzptq 1 “ σzptq9zptq ´ ´1σ `c dt k´1 2 2 N T k“1 ˆ ˙ N „ ÿ ? 1 γ T γ zk´1 pzk ´ zk´1 q ´ zk´1 ´ pzk ` zk´1 q ` ´1c “σ . 2 2 4 N k“1 The following lemma can be easily proved. Lemma 1. We have the following formula for c1 , c2 P C. ż

C

285

˙* " ˆ ˆ ˙ 1 1 1 2 σdz1 dz1 exp σ ´ |z|2 ` z1 z ´ c2 z ` c1 ` z1 2π 2 2 2 " ˆ ˆ ˙ ˙* z z 2 “ exp σ z . ` c1 ´ 2c2 ` c1 2 2

Using this lemma, we can compute the path integral explicitly as follows: KYp pz1 , z; T q ż ż σdzN´1 dzN´1 σdz1 dz1 ¨¨¨ “ lim ¨¨¨ NÑ8 C 2π 2π C # ˆ ˙ + N „ ÿ ? 1 γ γ T ˆ exp σ zk´1 pzk ´ zk´1 q ´ zk´1 ´ pzk ` zk´1 q ` ´1c 2 2 4 N k“1 ż ż σdzN´1 dzN´1 σdz1 dz1 ¨¨¨ “ lim ¨¨¨ NÑ8 C 2π 2π # C ˜ ¸ N ÿ 1 zk T γT 2 ´ |zk´1 | ` zk´1 p ´ ˆ exp σ q ` zk´1 2 2 2N 2N k“1

328

The Borel-Weil theorem and the Feynman path integral ? γT ´ ´1σcT `σpzN ´ z0 q 4N

329

+

ż σdzN´1 dzN´1 σdz1 dz1 ¨¨¨ ¨¨¨ NÑ8 C 2π 2π C " ˆ ˙ γT γT 1 2 ˆ exp σ ´ |z0 | ´ z0 ` z0 2 2N 2N ˆ ˆ ˆ ˙ ˙˙ z2 z0 γT 1 2 γT ` σ ´ |z1 | ` z1 ` z1 ´ ` 2 2 2N 2 2N ˆ ˆ ˙ ˙ z3 γT 1 2 γT ` σ ´ |z2 | ` z2 ` z2 ´ 2 2 2N 2N ˙ ˙ ˆ N ˆ ÿ γT γT 1 zk 2 ` zk´1 ´ ´ |zk´1 | ` zk´1 `σ 2 2 2N 2N k“4 * ? γT `σpzN ´ z0 q ´ ´1σcT 2N

“ lim

˙ ż σdzN´1 dzN´1 σdz2 dz2 ¨¨¨ ¨¨¨ NÑ8 C 2π 2π " ˆC ˙ 1 γT γT ˆ exp σ ´ |z0 | ´ z0 ` z0 2 2N 2N ˆ ˆ ˙ ˙ ˆ ˙˙ ˆ 1 2 γT z0 γT z3 z0 γT γT ` σ ´ |z2 | ` z2 ` z2 ´ ´ `2 ` 2 2 2N 2 2N N 2 2N ˆ ˙ ˙ N ˆ ÿ 1 zk γT γT ´ |zk´1 |2 ` zk´1 `σ ` zk´1 ´ 2 2 2N 2N k“4 * ? γT ` σpzN ´ z0 q ´ ´1σcT 4N “

ˆ

ż

lim

ż

repeating the above procedure,

286

ˆ ˙ ˆ ˙ ˙ " ˆ ? 1 2 z0 γT z0 1 γT “ lim exp σ ´ |z0 | ` zN ´ γT ´ ´1cT ` ` NÑ8 2 2 2 2 2 2 ˆ ˙* γT γT γT `σ ´z0 ` pz0 ´ zN q ` pz0 ` γT q 2N 4N 4N

329

330

Kiyosato Okamoto

˙* " ˆ ˙ ˆ ˙ ˆ ? 1 z γT z γT “ exp σ ´ |z|2 ` z1 ´ γT ´ ´1cT ` ` 2 2 2 2 4

¨

˛ 0 a c Thus for any Y “ ˝ 0 b‚ P g, we have 0 ż

σdzdz p 1 KY pz , z : T qFpzq C 2π " ˆ ż σdzdz 1 1 1 “ σ ´ |z|2 ` zpz ´ γT q ` z1 γT 2 2 2 C 2π ˙* ? 1 2 2 ´ |γ| T ´ ´1cT Fpzq 4 ˙* " ˆ ? 1 1 1 z γT ´ |γ|2 T 2 ´ ´1cT Fpz1 ´ γT q “ exp σ 2 4 “ pUλpσ pexp T YqFqpz1 q.

2 Borel-Weil theorem In this section, we consider unitary representations realized by the BorelWeil theorem for semisimple Lie groups. Let G be a connected semisimple Lie group such that there exists a complexification GC with π1 pGC q “ t1u and such that rank G “ dim T, T a maximal torus of G. Let K be a maximal compact subgroup of G which contains T , and k the Lie algebra of K. Note that G can be realized as a matrix group. We denote the conjugation of GC with respect to G, and that of gC with respect to g, both by - Let g and h be 287 the Lie algebras of G and T . We denote complexifications of g and h by gC and hC , respectively. Then hC is a Cartan subalgebra of gC . Let ∆ denote the set of all nonzero roots and ∆` the set of all positive roots. Then we have root space decomposition gC “ hC ` 330

ÿ

αP∆

gα .

The Borel-Weil theorem and the Feynman path integral Define n˘ “

ÿ

g˘α ,

331

b “ hC ` n´ .

αP∆`

Let N, N ´ , B and T C be the analytic subgroups corresponding to n` , n´ , b, and hC , respectively. We fix an integral form Λ on hC . Then exp H ÞÝÑ eΛpHq

ξΛ : T ÝÑ Up1q,

define a unitary character of T And ξΛ extends uniquely to a holomoprhic one-dimensional representation of B: ξΛ : B “ T C N ´ ÝÑ Cˆ ,

exp H ¨ n´ ÞÝÑ eΛpHq .

Let L˜ Λ be the holomorphic line bundle over GC {B associated to the holomorphic one-dimensional representation ξΛ of B. We denote by LΛ the restriction of L˜ Λ to the open submanifold G{T of GC {B: G

G{T

/ GB

/ GC

GB{B

“

/ GC {B

and LΛ

G{T

/ L˜ Λ

/ GC {B.

Then we can indentify the space of all holomorphic sections of LΛ with ! ) hol ´1 ΓpLΛ q “ f : GB ÝÑ C; f pxbq “ ξΛ pbq f pxq, x P GB, b P B . Let πΛ be a representation of G on ΓpLΛ q defined by πΛ pgq f pxq “ f pg´1 xq

for g P G, x P GB and f P ΓpLΛ q. 331

332

Kiyosato Okamoto

For any f P ΓpLΛ q we define

288 2

|| f || “

ż

G

| f pgq|2 dg,

where dg is the Haar measure on G. We put Γ2 pLΛ q “ t f P ΓpLΛ q; || f || ă `8u. Then the Borel-Weil theorem asserts that pπΛ , Γ2 pLΛ qq is an irreducible unitary representations of G (Bott [1], Kostant [8] and Harish-Chandra [2] [3] [4]). For the moment we assume that G is noncompact. We fix a Cartan decomposition of g: g “ k ` p. We denote complexification of k and p by kC and f pC , respectively. Let ∆c and ∆n denote the set of all compact roots and noncompact roots, respectively. Now we assume that Γ2 pLΛ q ‰ 0. Then there exists an ordering in the dual space of hR “ ih so that every positive noncompact root os larger than every compact positive root. The ordering determines ` sets of compact positive roots ∆` c and noncompact positive roots ∆n . Furthermore Λ satisfies the following two conditions: xΛ, αy ě 0

xΛ ` ρ, αy ă 0

for α P ∆` c ,

for α P ∆` n,

ř where ρ “ 21 αP∆` α. Then there exists a unique element ψΛ in ΓpLΛ q which satisfies the following conditions: πΛ phqψΛ “ ξΛ phqψΛ

dπΛ pXqψΛ “ 0

ψΛ peq “ 1, 332

for h P T,

for X P n` ,

The Borel-Weil theorem and the Feynman path integral

333

where dπΛ is the complexfication of the differential representation of πΛ . One ş can show that ψΛ is an element of ΓpLΛ q. We normalize dg so that G |ψΛ pgq|2 dg “ 1. Define D to be an open subset n` which satisfies exp D ¨ B “ GB X NB, where exp is the exponential map of n` onto N: „

exp : n` Y D

/N

Y / exp D.

For each α P ∆, we choose an Eα of gα such that

289

BpEα , E´α q “ 1 and

? ´1pEα ` E´α q P gu , ? where Bp¨, ¨q is the Killing form of gC and gu “ k ` ´1p, the compact real form of gC . Note that " ´E´α for α P ∆c , Eα “ E´α for α P ∆n Eα ´ E´α ,

We put m “ dim n` and introduce holomorphic coordinate on n` and n´ by ÿ zα E α , Cm Ñ n` , pzα qαP∆` ÞÝÑ z “ αP∆`

m

´

C Ñn ,

pwα qαP∆` ÞÝÑ w “

ÿ

αP∆`

We put nz “ exp n´ w “ exp

ÿ

αP∆`

ÿ

αP∆`

zα Eα P N, wα E´α P N ´ .

333

wα E´α

334

Kiyosato Okamoto

Let ΓpDq be the space of all holomorphic functions on D. The following correspondence gives an isomorphism of ΓpLΛ q into ΓpDq: Φ : ΓpLΛ q ÝÑ ΓpDq,

f ÞÝÑ F,

where Fpzq “ f pnz q for z P D. We put HΛ “ ΦpΓ2 pLΛ qq. Let us denote by UΛ pgq the representation of G on HΛ such that the diagram Γ2 pLΛ q πΛ pgq

/ HΛ

Γ2 pLΛ q

UΛ pgq

/ HΛ

is commutative for all g P G. We normalize the invariant measure µ onG{T such that ˆż ˙ ż ż f pghqdh dµpgT q for any f P Cc8 pGq, f pgqdg “ G

290

G{T

T

ş where dh is the Haar measure on T such that T dh “ 1. We denote the measure on D also by µ which is induced by the complex analytic isomorphism: φ : D ãÑ G{T. By the definition of D, φpDq is open dense in G{T . For any x P NT C N ´ we denote the N´, T C ´ and N ´ -component by npxq, hpxq and n´ pxq, respectively. Then, for any f P ΓpLΛ q, g P G and h P T we have | f pghq| “ | f pgq| and

|ξΛ phpghqq| “ |ξΛ phpgqq|.

This shows that | f pgq| and |ξΛ phpgqq| can be regraded as functions on G{T . We put JΛ pzq “ |ξΛ phpφpzqqq|´2 . 334

The Borel-Weil theorem and the Feynman path integral

335

Then we have ż

2

G

| f pgq| dg “ “

ż

G{T

ż

D

| f pgq|2 dµpgT q

|Fpzq|2 JΛ pzqdµpzq.

We define Γ2 pDq “ tF P ΓpDq; ||F|| ă `8u, where ||F||2 “

ż

D

|Fpzq|2 JΛ pzqdµpzq.

In case that G is compact, we remark in the above that G “ K,

GB ´ GC ,

∆c “ ∆,

D “ n` ,

∆n “ H,

g “ k,

p “ 0,

Γ2 pLΛ q “ ΓpLΛ q,

and ΓpLΛ q ‰ t0u if and only if Λ is dominant. For the rest of this paper we assume that Γ2 pLΛ q ‰ t0u. Suppose that G is noncompact. We put ÿ p˘ “ gα . αP∆˘ n

We denote by K C , P` and P´ be the analytic subgroups of GC corresponding to kC , p` and p´ , respectively. Then there is a unique open subset Ω of p` such that GB “ GK C P´ “ exp ΩK C P´ . We put W “ P` K C P´ . Then ψΛ is uniquely extended to a holomorphic function on W Henceforth, throughout the paper, the discussions are valid for the 291 compact case as well as for the noncompact case. Define a scalar function KΛ on GB ˆ GB by KΛ pg1 , g2 q “ ψΛ pg˚2 g1 q. Then KΛ p¨, g2 q, with g2 fixed, can be regarded as an elemetn of Γ2 pLΛ q. 335

336

Kiyosato Okamoto We define a scalar function KΛ on D ˆ D by KΛ pz1 , z2 q “ KΛ pnz1 , nz2 q.

Note that KΛ pz1 , z2 q is holomorphic in the first variable and anti-holomorphic in the second and that it can be regarded, with nz2 fixed, as an element of HΛ . Now we define operators KΛ and KΛ on Γ2 pLΛ q and HΛ by ż 2 KΛ pg2 , g1 q f pg1 qdg1 for f P Γ2 pLΛ q pKΛ f qpg q “ G

and 2

pKΛ Fqpz q “

ż

D

KΛ pz2 , z1 qFpz1 qJΛ pz1 qdµpz1 q for

F P HΛ ,

where dg1 is the Haar measure on G. Then we have the following commutative diagram: / HΛ Γ2 pLΛ q KΛ

KΛ

Γ2 pLΛ q

/ HΛ .

The important fact which we use in the next section is that KΛ is the indentity operator.

3 Path Integrals We keep the notation of the previous section. In [5], we tried to compute path integrals for the complex polarization of S Up1, 1q and S Up2q and we encountered the curcial difficulty of divergence of the path integrals. In [6], by taking the operator ordering into account and the regularizing the path integrals by use of the explicit form of the integrand, we computed path integrals and proved that the path integral gives the kernel fucntion of the irreducible unitary 292 representation of S Up1, 1q and S Up2q. In [7], we gave an idea how to 336

The Borel-Weil theorem and the Feynman path integral

337

regularize the path integrals for complex polarizations of any connected semisimple Lie group G which contains a comnpact Cartan subgroup T and showed, along this idea, that the path integral gives the kernel function of the irreducible unitary representation of G realized by Borel-Wiel theory. Our idea is, in a sense, nothing but to reularize the path integral using “normal ordering” (cf. Chapter 13 in [10]) and can be explined as follows. ? Put λ “ ´1Λ. We extend λ to an element of the dual space of g which vanishes on the orthgonal complement of h in g with respect to the killing form. Then for any element Y of the Lie algebra of G, the Hamiltonian on the flag manifold G{T is defined by HY pgq “ xAd˚ pgqλ, Yy ? “ ´1ΛpAdpg´1 qYq. Since the path integral of this Hamiltonian is divergent we regularize it by replacing e

?

´1HY pgq

“ eΛpHpAdpg

´1 qYqq

“ ξΛ pexppHpAdpg´1 qYqqq by ξΛ phpexppAdpg´1 qYqqq, whereH and h denote the projection operators: H : n` ` hC ` f n´ ÝÑ hC ,

h : exp n` exp hC exp n´ ÝÑ exp hC “ T C . Remark 1. For simplicity we assume that G is realized by a linear group. For any X P gC , we decompose X “ X` ` X0 ` X´ , where X` P n` , X0 P hC and x´ P n´ . We define the “normal ordering” : : by the rule such that the elements in n` appear in teh left, the elements in hC in the middle and the elements in n´ in the right. Then we have : exppX` ` X0 ` X´ q :“ exp X´ exp X0 exp x´ . 337

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Kiyosato Okamoto

For any X P gC we define ξΛ pexp Xq “ eΛpXq . Then the above regularization means that we replace ξΛ pexppX` ` X0 ` X´ qq 293

by ξΛ pexp x´ qξΛ pexp X0 qξΛ pexp X´ q. Before we start computing path integrals on the flag manifold G{T we prepare several lemmas. Lemma 2. For any g P NB we decompose it as g “ nz n´ wt

´ C where nz P N, n´ w P N ,t P T .

Suppose that g P G X NB. Then we can express w in terms of z and z which we denote w by wpz, zq. And we have KΛ pz, zq “ ξΛ pt˚ tq and

JΛ pzq “ KΛ pz, zq´1 .

For any z P D, we put gpz, zq “ nz n´ . wpz,zq Let d denote the exterior derivative on D. We decompose it as d “ B ` B, where B and B are holomorphic part and and anti-holomorphic part of d, respectively. Define ´1 ´ ´1 θ “ λpg´1 dgq “ λpn´1 w nz Bnz nw q ` λpt dtq,

where g “ nz n´ w t P G. And we choose ´1

´ α “ λpnw´ n´1 z Bnz nw q.

For any Y P f g, the Hamiltonian functions is given by HY pgq “ xAd˚ pgqλ, Yy “ xAd˚ pgpz, zqqλ, Yy, where g “ gpz, zq P G. 338

339

The Borel-Weil theorem and the Feynman path integral

If we decompose g “ nz n´ w t P G X NB as in Lemma 2, then we can show that ´1

´ ˚ ´1 ˚ Λpnw´ n´1 z Bnz nw q “ ´Λpptt q Bptt qq

“ ´B log KΛ pz, zq.

? It follows that α “ ´ ´1B log KΛ pz, zq. Now we consider the Hamiltonian part of the action. Let Kw pzq “ KΛ pz, wq and regard it as an element of HΛ . For any X P gC , we decompose it as X “ X` `H`X´ with X˘ P n˘ 294 and H P hC . Then we put HpXq “ H.

Lemma 3. For any X P gC and g “ nz n´ w t P G X NB, using the above notation, we have ξΛ phpexp εXqq “ ξΛ pexp εHpXqq ` Opεq2 and pUΛ pexp εXqKz qpzq “ Kz pzqξΛ phpg´1 exp εXgqq,

for sufficiently small ε. We put z0 “ z and zN “ z1 . First we compute the path integrals without Hamiltonians. Taking the same paths as in [5], we generalize Propositions 6.1 and 6.2 in [5] as follows: ˆ ˙ żT ż ? ˚ ´1 γ α Dpz, zq exp 0 ˜ ż k ¸ ż N´1 ż N ź ÿ NT B log KΛ pzptq, zk´1 q ¨¨¨ “ lim dµpzi q exp NÑ8 D

“ lim

ż

NÑ8 D

“ lim

ż

NÑ8 D

¨¨¨ ¨¨¨

D i“1 ż N´1 ź

˜

k“1 N ÿ

k´1 N T

D i“1

KΛ pzk , zk´1 q dµpzi q exp log KΛ pzk´1 , zk´1 q k“1

D i“1

dµpzi q exp

ż N´1 ź

N ź KΛ pzk , zk´1 q K pz , z q k“1 Λ k´1 k´1

339

¸

340

Kiyosato Okamoto

“ lim

ż

NÑ8 D

¨¨¨

ż

D

N´1 ź

JΛ pz0 q

i“1

“ JΛ KΛ pz1 , zq,

JΛ pzi qdµpzi q

N ź

k“1

KΛ pzk , zk´1 q

where we used Lemma 2 and the fact that KΛ is the identity operator. Next, for any Y P g, we quantize the Hamiltonian HY by choosing the following ordering: z Ñ zk´1 .

z Ñ zk ,

(C1 )

In [5] we proposed to compute the path integral in the following way: ż

ˆ

? ´1

żT

˙

˚

γ α ´ HY pgpz, zqqdt ¸ ˜ ż N´1 ż N ź ÿ KΛ pzk , zk´1 q ¨¨¨ “ lim dµpzi q exp log NÑ8 D KΛ pzk´1 , zk´1 q D i“1 k“1 ˜ ¸ N ÿ T ˆ exp ΛpAdpgpzk , zk´1 qq´1 Yq . N k“1

295

Dpz, zq exp

0

However this integral diverges. Therefore we replace eΛpAdpgpzk ,zk´1 qq

´1 Yq T N

“ ξΛ pexp Hp

T Adpgpzk , zk´1 qq´1 Yqq N

by ξΛ phpexpp

T Adpgpzk , zk´1 qq´1 Yqqq. N

(C2 )

Then our path integral, which generalizes the path integral given in [6], becomes lim

NÑ8

ż

D¨¨¨

ż

D

JΛ pz0 q

ˆ ξΛ phpexpp

N´1 ź i“1

JΛ pzi qdµpzi q

N ź

k“1

KΛ pzk mzk´1 q

T Adpgpzk , zk´1 q´1 qYqqq. N 340

The Borel-Weil theorem and the Feynman path integral

341

By Lemma 3, we see that KΛ pz1 , z2 qξΛ phpgpz1 , z2 q´1 exp

T Ygpz1 , z2 qqq N

is extended to the function T ψΛ pn˚z2 expp´ Yqnz1 q N defined on D ˆ D which is holomorphic in z1 and anti-holomorphic in z2 . To proceed further, we need the following Lemma 4. For any X P g and g1 , g2 P GB, ż ψΛ pn˚z g2 qψΛ pg1˚ exp Xnz qJΛ pzqdµpzq D ż ψΛ pn˚z exp Xg2 qψΛ pg1 nz qJΛ pzqdµpzq. “ D

2 1 Applying this `lemma ˘ to the path integral by taking X, g and g in Lemma T T 4 as - N Y, exp N Y nzk and nzk´1 for each k, respectively, we see that the path integral equals

JΛ pzqψΛ pn˚z expp´T Yqnz1 q

“ JΛ pzqKΛ pexpp´T Yqnz1 , nz q.

Furthermore, we have ż JΛ pzqKλ pexpp´T Yqnz1 , nz qFpzqdµpzq “ pUΛ pexppT YqqFqpz1 q D

for any F P HΛ . Thus we have obtained the following theorem. Theorem. For any Y P g, choosing the ordering (C1 ) and taking the regularzation (C2 ), the path integral of the Hamiltonian HY gives the kernel function of the operator UΛ pexppT Yqq. Remark 2. In case that G is compact,the theorem is valid for any Y P gC , because Lemma 4 holds for any X P gC in this case. 341

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Kiyosato Okamoto

References [1] R. Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957) 203–248. [2] Harish-Chandra, Representaions of semisimple Lie groups IV, Amer. Math. J. 77 (1955) 742–777. [3] Harish-Chandra, Representations of semisimple Lie groups V, Amer. Math. J. 78 (1956) 1–41. [4] Harish-Chandra, Representations of semisimple Lie groups VI, Amer. Math. J. 78 (1956) 564–628. [5] T. Hashimoto, K.Ogura, K.Okamoto, R.Sawae and H. Yasunaga, Kirillov-Kostant theory and Feynman path integrals on coadjoint orbits I, Hokkaido Math. J. 20(1991) 353–405. [6] T.Hashimoto, K.Ogura, K.Okamoto, R.Sawae, Kirillov-Kostant theory and Feynman path integrals on coadjoint orbits of S Up2q and S Up1, 1q, to appear in Proceedings of the RIMS Research Project 91 on Infinit Analysis. [7] T.Hashimoto, K.Ogura, K.Okamoto, R.Sawae,Borel-Weil theory and Feynman path integrals on flag manifolds, Hiroshima Math. J. (to appear). 297

[8] B.Kostant, Lie algebra cohimology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961) 329–387. [9] M.S. Narasimhan and K. Okamoto,An analogue of the Borel-WeilBott theorem for hermitian symmetric pairs of non-compact type, Ann. of Math. 91 (1970) 486–511. [10] A. Pressley and G. Segal, Loop Groups, Clarendon Press, Oxford, 1986. [11] H. Yasunaga, Kirillov-Kostant theory and Feynman path integrals on coadjoint orbits II, Hiroshima Math. J. (to appear). 342

The Borel-Weil theorem and the Feynman path integral Department of Mathematics Faculty of Science Hiroshima University

343

343

Geometric Super-rigidity Yum-Tong Siu*

Introduction A more descriptive title for this talk should be: “The superrigidity of Margulis as a consequence of the nonlinear Matsushima vanishing theorem”. What is presented in this talk is the culmination of an investigation in the theoey of geometric superrigidity which Sai-Kee-Yeung and I started about two years ago. We first used the method of averaging and invariants to obtain Bochner type formulas which yield geometrix superrigidity for the Grassmannians and some other cases. Finally we obtained a general Bochner type formula which includes the usual formulas of Bochner, Kodaira, Matsushima, and Corlette as well as those obtained by averaging so that all cases of geometric superrigidity in its most general form can be derived from such a general Bochner type formula I would like to point out that, for the difficult cases such as those with a Grassmannian of rank at least two as domain and a Riemannian manifold wity nonpositive sectional curvature as target, the formula form the Matsushima vanishing theorem does not yield geometric supperigidity. For those difficult cases one needs the cases of the general Bochner type formula motivated by the method of averaging and invariants. Even with the other simpler 300 cases for which the formula from the Matsushima vanishing theorem yields geometric superrigidity, to get the result with only the assumption of nonnegative sectional curvature for the target manifold instead of the stronger assumption of nonnegative curvature operator condition, one needs the use of an averaging argument. 299

* Partially

supported by a grant from the National Science Foundation

344

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Geometric Super-rigidity

Geometric superrigidity means the Archimedian case of Margulis’s superrigidity [?] formulated geometrically by assuming the target manifold to be only a Riemannian manifold with nonpositive curvature condition instead of locally symmetric. The complex case of Mostow’s strong rigidity theorem [Mos] is a consequence of the nonlinear version of Kodaira’s vanishing theorem which yields a stronger result requiring only the target manifold to be suitably nonpositively curved rather than locally symmetric [Si]. It turns out that in the same way the Archimedian case of Margulis’s superrigidity is a consequence of the nonlinear version of Matsushima’s vanishing theorem for the first Betti number [Mat]. Again the result is stronger in that the target manifold is required only to be suitably nonpositively curved instead of locally symmetric. Moreover, this approach provides a common platform for Margulis’s supperrigidity for the case of rank at least two and the recentsupperigiduty result of Corlette [Co] for the hyperbolic spaces of the quaternions and the Cayley numbers. The reason for the such vanishing theorem is the holonomy group which explians why supperigidity works for rank at least two as well as the hyperbolic spaces of quaternions and the Cayley numbers. The curvature RpX, Yq as an element of the Lie algebra of EndpT M q generates the Lie algebra of the holonomy group. The minimum condition is that the holonomy group is Opnq which simply says that RpX, Yq is skew-symmetric. To get a useful vanishing theorem one needs an additional condition to remove a term involving only the curvature of the domain manifold. The K¨ahler case is the same as the holonomy group being Upnq. Then RpX, Yq is C-linear as an element of EndpT m q. This additional condition enables one to obtain the Kodaira vanishing theorem for negative line bundles. Other holonomy groups help yield vanishing theorems for geometric superrgidity. One can also get vanishing theorems for some of the special holonomy groups. The approach to geometric superrigidity as the nonlinear version of Matsushima’s vanishing theorem is motivated by a remark which E. Calabi made privately to me during the Arbeitstagung of 1981 when I delivered a lecture on the newly discovered approach to the complex case of Mostow’s strong rigidity as the nonlinear version of Kodaira’s vanishing theorem. Calabi remarked that there is another vanishing theorem, 345

346

Yum-Tong Siu

namely Matsushima’s which one should look at. He also remarked that Kodaira’s vanishing theorem involves the curvature tensor quadratically [Ca]. Actually the early rigidity result of A. Weil [W] already depends 301 on Calabi’s idea of integrating the square of the curvature [Mat, p. 316] and this early rigidity result launched the theory of strong rigidity and superrigidity. ¿From this point of view it is not surprising that superrdidity cane be approached from Matsushima’s vanishing theorem. We state first here the final result we obtained. Theorem 1. Let M be a compact locally symmetric irreducible Riemanninan manifold of nonpositive curvature whose universal cover is not the real or complex hypebolic space. Let N be a Riemannian manifold whose complexified sectional curvature is nonpositive. If f is a nonconstant harmonic map from M to N, then the map from the universal cover of M to that of N induced by f is a totally geodesic isometric embedding. Here nonpositive complexified sectional curvature means that RN pV, W; V, Wq ď 0

for any complexified tangent vectors V, W at any xǫN, where RN is the curvature tensor of N. In this talk we follow the convention in Matusushima’s paper [Mat] that Ri ji j is negative for a negative curvture tensor [[Mat], p. 314, line 6]. Theorem 2. In Theorem 1 when the rank of M is at least two, one can replace the curvature condition of N by the weaker condition that the Riemananian sectional curvature of N is nonpositive. When the universal cover of M is bounded symmetric domain of rank at least two, Theorem 1 was proved by Mok [Mo]. When the universal cover of M is the hyperbolic space of the quaternions and the Cayley numbers, Corlette’s result differs from Theorem 1 only in that Corlette’s result requires the stronger curvature condition that the quadratic form pξi j q ÞÑ RiNjkl ξi j ξkl be nonpositive for skew-symmetric pξi j q. Theorem 3. In Theorems 1 and 2 let X be the universal cover of M and Γ be the fundamental group of M. Then the conclusions of Theorems1 346

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Geometric Super-rigidity

and 2 remain true when the harmonic map f from M to N is replaced by a Γ-equivariant harmonic map f from X to N. Remark. With the existence result for equivariant harmonic maps corresponding to the results of Eells-Sampson [E-S], Theorem 3 implies the following Archimedian case of the superrigidity theorem of Marugulis [?]: For lattices Γ and Γ1 extends to a homomorphism from G to G1 , when G is noncompact simple of rank at least two and Γ is cocompact. The general Archimedian case of the superrigidity theorem of Margulis 302 would follow from the corresponding generalization of Theorem 3. In order not to distract from the key points of our arguments, we will not discuss such generalizations in this talk. Also we will focus only on Theorems 1and 2, because the modifications in the proofs of Theorems 1 and 2 needed to get Theorem 3 are straightforward.

An Earlier Approach of Averaging After Corlette [Co] obtained the superrigidity for the case of the hyperbolic spaces of the quaternions and the Cayley numbers, Sai-Kee Yeung and I started to try to undersatand how Corlette’s result could be fitted in a more complete global picture of geometric superrigidity. Corelette’s method is to generalize the method of the nonlinear BB-Bochner formula for the complex strong rigidity by replacing the K¨ahler form used there by the invariant 4-form in the case of the quaternionic hyperbolic space. That 4-form corresponds to the once on the quaternionic projective space whose restriction to a quaternionic line is its standard volume form. Later Gromov [G] introduced the method of foliated harmonic maps so that Corlette’s result could be proved by applying the nonlinear BB-Bochner formula to the leaves. In his proof of the case of Theorem1 when the universal cover of M is a bounded symmetric domain of rank at least two, Mok [Mo] remarked that, according to Gromov, one should be able to develop the foliation technique of Gromov [G] to extend Mok’s proof to many Riemannian symmetric manifolds of the noncompact type with rank at least two by considering families of totally geodesic Hermitian symmetric submanifolds of rank at least two. 347

348

Yum-Tong Siu

The earlier approach Sai-Kee Yeung and I adopted was motivated by Gromov’s work on foliated harmonic maps. We started out by considering a totally geodesic Hermitian suymmetric submanifold σ of the universal cover X of M. We look at the nonlinear BB-Bochner formula applied to the restriction of the Hermitian-symmetric submanifold σ and the average over all such submanifolds under the action of the automorphism group of X. More precisely, we let X be the quotient of a Lie group G by a maximum compact subgroup K and let Γ be the fundamental group of M. Choose a suitable subgroup H of G so that H{pA X Kq is a bounded symmetric domain of complex dimension at least two. We pull back the harmonic map f : M Ñ N to a map f˜ from G{K to N and, for every k is K, apply the nonlinear BB-Bochner technique developed in [[Si], [Sa]] to the restriction of f˜ to k ¨ pH{pH X Kqq. Since the image of k ¨ pH{pH X Kqq in ΓzX is noncompact, one has to introduce a method 303 a averaging over k to handle the step of integration by parts. As a result of averaging over k the integrand of the gradient square term of the differential of the map f in the formula is an averaged expression of the Hessian of f . The difficult step in this approach is to determine under what condition this averaged expression of the Hessian of f is positive definite in the case of a harmonic map. It turns out that in some cases when we use only one single subgroup H of G this averaging expression in general is not positive definite for harmonic maps. To overcome this difficulty we choose two subgroups H1 and H2 instead of a single H and we sum the BB-Bochner formulas for the two subgroups. For example, this is done in the case of S Opp, qq{S pOppq ˆ Opqqq for p ą 2 and q ą 2 pν “ 1, 2q and the sum of the two averaged expressions of the Hessian of f turns out to be positive definite for harmonic maps for this case. In Cartan’s classification of Riemannian symmetric manifolds, besides the ten exceptional ones there are only the following four series which are not Hermitian symmetric:S Opp, qq{S pOppq ˆ Opqqq, S ppp, qq{S pppq ˆ S ppqq, sUpkq{S Opkq, and S U ˚ p2nq{S ppnq. We explicitly verified that for these four series the averaged expression of the Hessian of the Hessian of f is positive definite in the case of a harmonic 348

349

Geometric Super-rigidity

map so that both Theorem 1 and Theorem 2 hold for these four series. The method of verification is to use scalar invariants from the representation of compact groups and Cramer’s rule. More precisely, let V be a finite-dimensional vector space over R with an inner product ă ¨, ¨ ą. Let K be a compact subgroup of the special orthogonal group S OpVq with respect to the inner product. Let S be an element ş ‘4 of V . To compute the average gPK g ¨S , we first enumerate all the ‘4 şone-dimensional řk K-invariant subspaces RIκ p1 ď κ ď kq of V so that κ“1 cκ Iκ for some constants cκ . By taking the inner prodgPkg ¨S “ uct of this equation with IΛ , we have the system of linear equations řk κ“1 cκ ă Iκ , IΛ ą“ă S , IΛ ą from which we can use Cramer’s rule to solve for the constants cκ . For such verification it does not matter whether one uses the original Riemannian symmetric space or its compact dual and we will use its compact dual in the following description of the verification. For the case of G “ S Opp, qq and K “ S pOppq ˆ Opqqq for p ą 2 and q ą 2 we use the two subgroups H1 “ S Opp, 2q and H2 “ S Op2, qq of G so that H j {pH j X Kq is a bounded symmetric domain of rank two. The tangent space of G{K is given by a p ˆ q matrix and we denote the second partial derivative of the map f with the pα, βqth entry and the pγ, δqth entry by fαβ,γδ . (Similar notations are also used for the description of the other three seres without further explanation.) Then the avearaged expression Φσ1 of the Hessian of f for the subgroup 304 H1 “ S Opp, 2q is ˆ ˆ ˙ 1 2 Φσ1 “ fαβ,αβ fγδ,γδ ` 1 ´ fαβ,αδ fγβ,γδ q pq ´ 1qpq ` 2q ˙ 2 ´ fαβ,γβ fαδ,γδ ´ fαβ,γδ fαβ,γδ ` fαβ,γδ fαδ,γβ , q where the summation convention of summing over repeated indices is 1 1 Φσ1 ` qpq´1q Φσ1 is positive definite when minpp, qq used. Moreover, ppp´1q ě 3. The expression Φσ j (and also similar expressions lates) is given only up to a positive constant depending on the total measure of th compact group K. 349

350

Yum-Tong Siu

For the case of G “ S ppp, qq and K “ S pppq ˆ S ppqq, the totally geodesic Hermitian symmetric submanifold usedˆis S Upp`qq{spUppqˆ ˙ c D Upqqq. The tangent space of G{K is the set of . Before we ´D C ˆ ˙ C D average, we lift the expression with arguments in to an ex´D C pression with arguments in a general pp`qqˆppˆqq matrix W “ pwαi q B2 f so that with the notation Bαi Bβ j f “ we have the symmetry Bwαi Bwβ j B JpβqJp jq B jpαqJpiq f “ Bαi Bβ j f , where Jpαq “ p ` α and Jpp ` αq “ ´α with Bp´αqi meaning ´Bαi . The averaged expression of the Hessian of f is 3p ´ pp ` 2q| fαiβ j |2 ´ p2p ` 1q fαiβ j fαpJ qβpJiq f f j 2 αiαi γ jγ j for q “ 1 and is pp ` q ` 2pqq fαiαi fβ jβ j ´ p1 ` pq fαiα j fβ jβi

´p1 ` qq fαiβi fβ jα j ´ pp ` q ` 2pqq fαiβ j fβ jαi ´p2 ` p ` qq fαiβ j fβpJiqαpJ jq

for q ą 1 and is positive definite when minpp, qq ě 1 and maxpp, qq ě 2. For the case of G “ S Lpk, Rq and K “ S Opkq, we let n be the largest integer with 2n ă k. The totally geodesic Hermitian symmetric submanifold used is S ppnq{Upnq. The tangent space of G{K is the set of all symmetric matrices of order k with zero trace. The averaged expression of the Hessian of f is equal to fαβ,αβ fλµ,λµ ´

4 4 fαβ,γβ fαµ,γµ ´ fαβ,γδ fαβ,γδ ` fαβ,γδ fαγ,βγ k`2 k`2

which is nonnegative for k ě 4. For the case of G “ S Up2nq and K “ S ppnq, the Hermitian symmetric submaifold is S Op2nq{Upnq. The tangent space of G{K is given 350

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Geometric Super-rigidity 305

by the set of all px, ˆ Yq of the ˙ form pX, Yq “ pA ´ D, B ` Cq with the A B p2nq ˆ p2nq matrix skew-Hermitian and tracesless. Befor we C D average, we lift the second derivative of f , via the map Z “ pzαβ q “ ˆ ˙ A B ÞÑ pA ´ D, B ` Cq to the second jet fαβδγ “ Bzαβ Bzγδ f on C D the Lie algebra of S Up2nq so that the symmetries fαβδγ “ ´ fβαδγ and f JpβqJpαqδγ “ fαβδγ hold, where as earlier Jpαq “ n ` α and Jpn ` αq “ ´α. The averaged expression of the Hessian of f is fαββα fγδδγ ´

2 2 fαβγα fβδδγ ´ fαβγδ fβαδγ ´ f fδαβγ n´1 n ´ 1 αβγδ

which is nonnegative for n ě 3. In the above approach by averaging, the natural curvature condition for the target manifold is the nonnegativity of the complexified sectional curvature. One can also consider the curvature term obtained by averaging and argue by the number of invariants that the target manifold needs only to satisfy the weaker condition of the nonnegativity of the sectional curvature in the case of the domain manifold of rank at least two. Mok came up with the the idea that to get directly the weaker condition of nonnegative sectional curvature for the target manifold, one can restrict the harmonic map to totally geodesic flat submanifolds of the domain manifold and average the usual nonlinear Bochner formula there instead of the nonlinear BB-Bochner formula. Though this averaging method theoretically can also be applied to the ten exceptional cases of Riemannian symmetric manifolds which are not Hermitian symmetric, explicit computation becomes cumbersome for them. We then changed our approach and used instead the nonlinear Matsushima vanishing theorem in our investigations of the ten exceptional cases. The use of the nonlinear Matsushima vanishing theorem in the exceptional cases is the most natural approach. In the course of our investigation involving both the Bochner type formula from averaging and those from the Matsushima vanishing theorem we came to a much better understanding of the nature of such vanishing theorems. We could formulate such vanishing theorems in a general setting. The most 351

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general case of geometric superrigidity is then a consequence of such a general nonlinear vanishing theorem. Both the BB-Bochner vanishing theorem and the Matsushim vanishing theorem are special cases of teh vanishing theorem for the general setting. It also gives a very short and elegant proof of the original Matsushima vanishing theorem.

Matsushima’s Vanishing Theorem 306

Matsushima’s theorem states that the first Betti number of a compact complex manifold is zero if its universal cover is an irreducible bounded symmetric domain of rank at least two. One step of Matsushima’s original proof is the verification of the positivity of a certain quadratic form ÿ ÿ pξi j q ÞÑ bpgq pξi j q2 ` Rikh j ξi j ξkh , i, j

i, j,k,l

where bpgq is a constant depending on and explicitly computable from the Lie algebra g of the Hermitian symmetric manifold and Rikh j is the curvature tensor of the Hermitian symmetric manifold. The verification makes use of the computations by Calabi-Vesentini and Borel on the eigenvalues of the quadratic form given by the curvature tensor acting on the symmetric 2-tensors of a Hermitian symmetric manifold. Mostow’s strong rigidity theorem (for the case of simple groups) says that if G and G2 are noncompact simple groups not equal to PS L p2, Rq and Γ Ă G and Γ1 Ă G1 are lattices, then any isomorphism can be extended to an isomorphishm from G to G1 . For the case of bounded symmetric domains and cocompact lattices we can state it as follows. Let D and D1 be irreducible bounded symmetric domains of complex dimension at least two and let M and M 1 be respectively smooth compact quotients of D and D1 . If M and M 1 are of the same homotopy type, then M and M 1 are biholomorphic (or anti-biholomorphic). The vanishing theorem of Kodaira for a negative line bundle L over a compact K¨ahler manifold M of complex dimension n ě 2 can be proved as follows. We do it for the vanishing of H 1 pM, Lq becaucse that is the 352

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case we need. Suppose ξ is an L-valued harmonic p0, 1q-form on M. Let ω be a K¨ahler form of M. Then ż ż ? ? n´2 2 “ ||Dξ|| ´ pRL ξ, ξq 0“ ´1BBp ´1ξ ^ ξq ^ ω M

M

implies that ξ vanishes, where RL is the curvature of L. The nonlinear version of Kodaira’s vanishing theorem is as follows. Let M and N be compact K¨ahler manifolds and f : M Ñ N be a harmonic map which is a homotopy equivalence. Use B f instead of ξ. We get ż ? ? 0“ ´1BBp ´1hαβ β f α ^ B f β q ^ ωn´2 “ ||DB f ||2 M ż RN pB f, B f, B f, B f q. ´ M

Suitable nonpositive curvature property of N implies that either B f or B f 307 vanishes. Such a curvature property is satisfied by irreducible bounded symmetric domains of complex dimension at least two. This nonlinear version implies the complex case pf Mostow’s strong rigidity theorem,because the theorem of Eells-Sampson implies the existence of a harmonic map in the homotopy class of continuous maps from a compact Riemannian manifold to a nonpositively curved Riemannian manifold. Moreover, the target manifold is assumed to satisfy only a curvature condtion instead of being locally symmetric. The complex case of strong rigidity corresponds to the vanishing of the first cohomology wiht coefficient in a coherent analytic sheaf. The real analog corresponds to the vanishing of the first cohomology with coefficient in the constant sheaf. So we should look at the vanishing of the first Betti number. On the other hand holomorphic means B “ 0. Its real analog should mean d “ 0 which means parallelism. The pullback of the metric tensor being parallel means isometry after renormalization. This consideration gives the motivation that the nonlinear version of Matsushima’s vanishing theorem for the first Betti number would yield the Archimedian case of Margulis’s superrigidity theorem 353

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with the assumption on the target manifold weakened from local symmetry to suitable nonpositive curvature. The reason for geometric superrigidity turns out be the holonomy group. the curvature RpX, Yq as an element of the Lie algebra of EndpT q generates the Lie algebra of the holonomy group. The minimum condition is Opnq which simply says that RpX, Yq is skew-symmetric. The K¨ahler case is the same as the holonomy group being Upnq. Then RpX, Yq is C-linear as an element of EndpT q. It is the same as saying that Rαβi j “ 0 for 1 ď alpha, β ď n and i, j running through 1, ¨ ¨ ¨ , n and 1, ¨ ¨ ¨ , n. The condition is equivalent to Rαβγδ being symmetric in α and γ by the Bianchi identity Rαβγδ ` Rαγδβ ` Rαδβγ “ 0.

Vahishing Theorems from 4-Tensors A vanishing theorem is the result of a 4-tensor Q satisfying the following conditions. This 4-tensor Qi jkl should be skew-symmetric in i and j and symmetric in pi, jq and pk, lq. Moreover, the following three conditions should be satisfied: ř (i) The quadratic form i, j,k,l Qi jkl ξil ξ jk is positive definite on all traceless ξi j . (ii) ă Ap¨, ¨, ¨, Xq, Rp¨, ¨, ¨, Yq ą“ 0 for all X, Y. 308

(iii) Q is parallel. Once one has such a 4-tensor Q, one applies integration by parts to ż Qi jkl ∇i fl ∇ j fk M

for any harmonic f to show that f is zero. Here ∇ denotes covariant differentiation. We can do this for the linear as well as the nonlinear version of the vanishing theorem. As and example let us look at Kodaira’s vanishing theorem. The 4-tensor is Qαβγ,δ “ δαδ δβγ ´ δαβ δγδ . 354

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Note that this Q is simply the curvature tensor for the manifold of consatant holomorphic curvature with the sign of the second term reversed. Then ¸˜ ˜ ¸ ÿ ÿ ÿ ÿ ξαδ ξαδ ´ Qi jkl ξil ξ jk “ Qαβγδ ξαδ ξβγ “ ξββ “ ξαα ξαδ ξα δ α

α,δ

β

α,δ

is positive definite. Moreover, Qi jkl Ri jkh “ Qαβγδ Rαβγh “ Rββδh ´ Rδββh “ 0. In the case of a harmonic 1-form f , the formula is simply ż ż Qi jkl fl ∇i ∇ j fk Qi jkl ∇i fl ∇ j fk “ ´ M M ż “ ‰ 1 “´ Qi jkl fl ∇i , ∇ j fk 2 M ż 1 “´ Qi jkl fl Ri jkh fh “ 0. 2 M Note that this gives a proof of Matsushima’s vanishing theorem when we consider a harmonic form fi , because the conditions on Q imply that fi is parallel and there is no nonzero parallel 1-form otherwise there is a deR-ham decomposition of the universal cover. In the case of a compact K¨a hler manifold (without using any line bundle or any map) applied to a harmonic p1, 0q-form fα the formula gives Bβ fα “ 0 for all α and β, which is the same as saying that any harmonic p1, 0q-form on a compact K¨ahler manifold is holomorphic. When this is applied to a harmonic 1form with values in a line bundle, we have another term in the formula represented by the curvature of the line bundle. Suppose the holonomy group is not Upnq. Then Berger’s theorem 309 forces then manifold to be locally symmetric except for the so-called exceptional holonomy groups. Assume that we have a compact locally symmetric manifold. Let K0 be the curvature tensor of constant curvature 1 given by pK0 qi jkl “ δik δ jl ´ δil δ jk . 355

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We are going to use Q “ c0 K0 ` R for some suitable constant c0 . The condition Qi jkl Ri jkh “ 0 simply says that ´2c0 Rl j jh ` Ri jkl Ri jkh “ 0. The factor Rl j jh in the first term is simply equal to the negative of the Ricci curvature pRicqlh according to the convention in Matsushima’s paper [Mat]. The second term Ri jkh Ri jkh is a symmetric 2-tensor which is parallel. Now every parallel symmetric 2-tensor is a constant multiple of the Kronecker delta, otherwise any proper eigenspace at a point would give rise to a deRham decomposition of the manifold. So we know that c0 exists. We can determine the actual value of c0 by contracting the indices h and l. We get c0 “ ´ ă R, R ą { ă R, K0 ą. Consider now the integration by parts of ż pc0 K0 ` Rqi jkl ∇i fl ∇ j fk . M

The question now is the positive definiteness of the quadratic form ξ ÞÑ pc0 K0 ` Rqi jkl ξil ξ jk on traceless ξ, which is the same as ÿ ÿ ξ ÞÑ c0 pξil q2 ` Ri jkl ξil ξ jk . i,l

(˚)

i, j,k,l

We now look at the nonlinear version. From r∇i , ∇k s fl we get an expression involving the curvature tensor of the target manifold. So ż ż Qi jkl fl ∇i ∇ j fk Qi jkl ∇i fl ∇ j fk “ M M ż 1 “ Qi jkl flD RNABCD fiA f jB fkC . 2 M To simplify notations we write p f ˚ RN qi jkl “ RNABCD fiA fiB fkC flD . So our final formula is ż ż 1 pc0 K0 ` Rqi jkl ∇i fl ∇ j fk “ ă c0 K0 ` R, f ˚ RN ą 2 M M 356

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It is simple and straightforward to verify that c0 ě bpgq. From the work of Kaneyuki-Nagano [K-N] we can conclude that the quadratic form (˚) 310 is positive definite. The Term Involving the Curvature of the Target Manifold. We have to worry about the sign of the term involving the curvature of the target ş manifold M ă c0 K0 ` R, f ˚ RN ą. We have to determine conditions on RN so that this term is nonpositive. We need only consider pointwise nonpositivity. Fix a point P0 of the domain manifold M. Let C be the vector space of all 4-tensors T i jkl which satisfies the following three symmetry conditions: (1)T i jkl “ ´T jikl , (2) T i jkl “ T kli j , and (3) T i jkl ` T ikl j ` T il jk “ 0. In other words, C is the vector space of all 4-tensors of curvature type. Let H denote the isotropy subgroup at that point. From the known results on the decomposition into irreducible representations of the representation of H on the skew-symmetric 2-tensors, we know that there are two, three, or four independent linear scalar H-invariants for elements of C. Consider first the case when there are only two independent linear scalar H-invariants given by inner products with the H-invariant elements Ii jkl and Ii1 jkl of C so that I “ K0 and ă I, I 1 ą“ 0. In our argument we can use either the complexified sectional curvature or the usual Riemannian sectional curvature (or even the analogously defined quaternionic or Cayley number sectional curvature). The arguments are strictly analogous. Let us assume that the rank of the domain manifold is at least two and consider the case of the usual Riemannian sectional curvature. Fix any 2-plane E in the tangent space of M at P0 so that the Riemannian sectional curvature SectpR, Eq of R for E is zero. Consider ş the following expression gPH Sectp f ˚ RN , g¨Eq. This expression is equal ` ˘ to a ă f ˚ RN , I ą `a1 ă f ˚ RN , I 1 ą for some real constants a and a1 depending on E.`On the other hand, the integrand˘ ă c0 K0 ` R, f ˚ R ą is of the form b ă f ˚ RN , I ą `b1 ă f ˚ RN , I 1 ą for some rea 1 constants b and b1 . Since both expressions vanish for f equal to the identity map, we conclude that b1 “ a1 . To compute a and a1 , we use K0 as the test value to replace f ˚ RN . The value b is given by b ă K0 , K0 ą“ c0 ă K0 , K0 ą ` ă R, K0 ą 357

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and the value of a is given by a ă K0 , K0 ą“ SectpK0 , Eq. Since c0 “ ´ ă R, R ą { ă R, K0 ą it follows from Schwarz’s inequality and the nonpositivity of ă R, K0 ą that b is nonnegative. From SectpK0 , Eq “ 1 we conclude that pc0 K0 ` Rqi jkl p f ˚ RN qi jkl is equal to ´ ă R, K0 ą´1 pă K0 , K0 ąă R, R ą ´ ă R, K0 ą2 q ż Sectp f ˚ RN , g ¨ Eq. gPK

311

We have thus the final formula ż pc0 K0 ` Rqi jkl ∇i fl ∇ j fk M

“ ´ ă R, K0 ą´1 pă K0 , K0 ąă R, R ą ´ ă R, K0 ą2 q ˆż ˙ ż f ˚ RN , g ¨ E P , PPM

gPHP

where E P is a 2-plane in the tangent space of M at P at which tthe Riemmanian sectional curvature of M is zero and HP is the isotropy group at P. So we have the geometric superrigidity result that any harmonic map from such a compact locally symmetric manifold to a Riemannian manifold with nonpositive Riemannian sectional curvature is a totally geodesic isometric embedding. The case of three or four independent linear scalar H-invariants occurs only in the case of Hermitian or quaternionic symmetric spaces or the case of Grassamanians. Let us illustrate the technique by looking at the Hermitian symmetric case. Let KC denote the curvature tensor of constant holomorphic sectional curvature. Instead of using Q “ c0 K0 ` R, one uses Q “ λpK0 ´ KC q ` µpc0 KC ` Rq for some suitable constants. This method of using a suitable linear combination is parallel 1 1 to the choice of the suitable constants ppp´1q , qpq´1q in the expression 1 ` qpq´1q Φσ2 in the earlier approach of averaging. Details of the methods and results described above will be in a paper to appear elsewhere. 1 Φ ppp´1q σ1

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References [B] A. Borel, On the curvature tensor of the hermitian symmetric manifolds, Ann. of Math. 71 (1960) 508–521. [Ca] E. Calabi, Matsushima’s theorem in Riemannian and K¨ahler geometry, Notices of the Amer. Math. Soc. 10 (1963) 505. [C-V] E. Calabi and E. Vesentini, On compact locally symmetric K¨ahler manifolds, Ann. of Math 71 (1960) 472–507. [Co] K. Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. 135 (1992) 165–182. [E-S] J. Eells and J. Sampson, Harmonic maps of Riemannian manifolds, Amer. J. Math. 86 (1964) 109–160. [G] M. Gromov, Foliated Plateau problem I II, Geom. and Funct. 312 Analy. 1 (1991) 14-79 and 253–320. [K-N] S. Kaneyuki and T. Nagano, it On certain quadratic forms related to symmetric Riemannian spaces, Osaka Math. J 14 (1962) 241– 252. [Mag] G.A. Margulis, Discrete groups of motoins of manifolds of nonpositive curvature, A. M.S. Transl. (1), 109 (1977) 33–45. [Mat] Y. Matsushima, On the first Betti number of compact quotient spaces of higher-dimensinal summetric spaces, Ann. of Math. 75 (1962) 312–330. ` Aspects of K¨ahler geometry on arithmetic varieties, Pro[Mo] N. M´ok, ceedings of the 1989 AMS Summer School on Several Complex Variables in Santa Cruz. [Mos] G. D. Motow, Strong Rigidity of Locally symmetric Spaces, Ann. Math. Studies 78, Princeton Univ. Press, (1973) 359

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[Sa] J. Sampson, Applications of harmonic maps to K¨ahler geometry, Contemporary Mathematics, Vol. 49 (1986) 125-133. [Si] Y.-T.Siu Complex-analyticity of harmonic maps, vanishing and Lefshetz theorems, J. Diff. Geom. 17 (1982) 55–138. [W] A. Weil, On discrete subgroups of Lie groups II, Ann. of Math. 75 (1962) 578–602.

Department of Mathematics Harvard University Cambridge, MA 02138 U.S.A.

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